From the question,
The two triangles that are congruent are :
Triangle CDM = Triangle ABM --------- Option C
What is the value of(-* - ) =( 4 )?-1516-516
Answer
Option B is correct.
[tex]-1\frac{5}{16}[/tex]Explanation
To answer this, we will first deal with the value in the first bracket by taking LCM
[tex]\begin{gathered} (-\frac{1}{4}-\frac{1}{2}) \\ =\frac{-1-2}{4} \\ =\frac{-3}{4} \end{gathered}[/tex]Then to solve the part with division, we know that the division involving fractions are solved by changing the division sign into multiplication sign and the fraction after the sign changes to its reciprocal or its inverse.
[tex]\begin{gathered} (-\frac{3}{4})\div\frac{4}{7} \\ =-\frac{3}{4}\times\frac{7}{4} \\ =-\frac{21}{16} \\ =-1\frac{5}{16} \end{gathered}[/tex]Hope this Helps!!!
Suppose a jar contains 6 red marbles and 27 blue marbles. If you reach in the jar and pull out 2 marbles at random at the same time, find the probability that both are red.
SOLUTION
Now the jar contains 6 red marbles and 27 blue marbles
Total number of marbles is
[tex]6+27=33\text{ marbles }[/tex]Now taking two red marbles at random means the first marble is red and the second marble is red.
Probability that the first marble is red is
[tex]\begin{gathered} =\frac{\text{ number of red marbles }}{\text{ total number of marbles}} \\ =\frac{6}{33} \end{gathered}[/tex]After taking the first red marble, we will have 5 red marbles remaining and a total of 32 marbles remaining
So probability of picking the second marble is
[tex]\begin{gathered} =\frac{\text{ number of red marbles remaining }}{\text{total number of marbles remaining }} \\ =\frac{5}{32} \end{gathered}[/tex]So probability both marbles are red means the first is red and the second is red.
And here means we have to multiply, this becomes
[tex]\begin{gathered} \frac{6}{33}\times\frac{5}{32} \\ =\frac{5}{176} \end{gathered}[/tex]Hence the answer is
[tex]\frac{5}{176}[/tex]Trigonometry Give the reference angle and the quadrant of the following
Answer:
To find the reference angle and the quadrant of,
[tex]675\degree[/tex]we have that,
Every angle is measured from the positive part of the x-axis to its terminal line traveling counterclockwise. If you want to find the reference angle, you have to find the smallest possible angle formed by the x-axis and the terminal line, going either clockwise or counterclockwise. If you want to find the reference angle, you have to find the smallest possible angle formed by the x-axis and the terminal line, going either clockwise or counterclockwise.
Since we get that,
The angle 675 degrees lies between (2x270=) 540 degrees and (2x360=) 720 degrees,
Therefore, the angle lies in the fourth quadrant.
To find the reference angle:
we get,
Reference angle is,
[tex]2\times360\degree-r=675\degree[/tex]where r is the reference angle.
Solving the above equation we get,
[tex]720\degree-r=675\degree[/tex][tex]r=720\degree-675\degree[/tex][tex]r=45\degree[/tex]The reference angle is 45 degrees and it lies in 4th quadrant.
If one zero of 5² + 13 + is the reciprocal of the other, find the value of k?
Answer:
k = 5
Explanation:
The given polynomial is
5² + 13 +
Let one of the zeros be z. Given that one of the zero is the reciprocal of the other, the reciprocal of z is 1/z. The roots are z and 1/z
The standard form of a quadratic polynomial is
ax^2 + bx + c
By comparing the polynomial expressions,
a = 5, b = 13, c = k
The product of the roots of a quadratic polynomial is c/a = k/5
Thus,
1/z * z = k/5
1 = k/5
By cross multiplying,
k = 5
the polynomial p(x)=2x^3+17x^2+41x+30 has a known factor of (x+5) rewrite p(x) as a product of linear functionp(x)=
Answer:
Rewriting p(x) as a product of linear functions, we have;
[tex]p(x)=(2x+3)(x+5)(x+2)[/tex]Explanation:
We want to write the given polynomial p(x) as a product of linear functions.
[tex]p(x)=2x^3+17x^2+41x+30[/tex]To write it as a product of linear functions we have to find the other factors;
let us divide the given polynomial by the given factor;
[tex]\begin{gathered} \text{ }2x^2+7x+6 \\ (x+5)\sqrt[]{2x^3+17x^2+41x+30} \\ \text{ - (}2x^3+10x^2) \\ \text{ }7x^2+41x+30 \\ \text{ }-(7x^2+35x) \\ \text{ }6x+30 \\ \text{ - (}6x+30) \\ \text{ 0} \end{gathered}[/tex]So, the division gives;
[tex]p(x)=2x^3+17x^2+41x+30=(x+5)(2x^2+7x+6)[/tex]next, we need to find the factors of the quadratic function;
[tex]\begin{gathered} 2x^2+7x+6 \\ 2x^2+4x+3x+6 \\ 2x(x+2)+3(x+2) \\ (2x+3)(x+2)_{} \end{gathered}[/tex]Substituting the factors of the quadratic function, we have;
[tex]\begin{gathered} p(x)=2x^3+17x^2+41x+30=(x+5)(2x^2+7x+6) \\ p(x)=(x+5)(2x+3)(x+2)_{} \\ p(x)=(2x+3)(x+5)(x+2) \end{gathered}[/tex]Therefore, rewriting p(x) as a product of linear functions, we have;
[tex]p(x)=(2x+3)(x+5)(x+2)[/tex]4. A field mouse hops along a parabolic path given by y=-0.2x^2+1.3x where x is the mouse's horizontal position (in feet) and y is the corresponding height (in feet).
Given that A field mouse hops along a parabolic path given by
[tex]y=-0.2x^2+1.3x[/tex]To find how far does the mouse jump, substitute with y = 0 and solve for x
so,
[tex]\begin{gathered} y=0 \\ -0.2x^2+1.3x=0 \\ x(-0.2x+1.3)=0 \\ x=0 \\ -0.2x+1.3=0\rightarrow x=\frac{1.3}{0.2}=6.5 \end{gathered}[/tex]So, the midpoint of the parabola will be:
[tex]x_m=\frac{6.5}{2}=3.25[/tex]so, substitute with the last value to find the highest point:
[tex]y=-0.2\cdot3.25^2+1.3\cdot3.25=2.1125[/tex]so, the answer will be y = 2.1125
This means the mouse can jump to 2.1125 feet
Also, the highest value of the bath will be 6.5 feet
Part B:
Can the mouse jump over 3 feet?
As the highest point is 2.1125 feet, the mouse can not jump over 3 feet.
Determine if the situation below are biased or unbiased and explain why. Two people from each 8th period class are asked what they think the theme of the next dance should be.
Given,
Two people from each 8th period class are asked what they think the theme of the next dance should be.
Required
The situation is biased and unbiased.
Here 2 students from each 8th class period is asking for dance theme.
The students are not getting equal chances.
So, the situation is unbiased.
Hence, the situation is unbiased.
Find a formula for the nth termof the arithmetic sequence.First term 9Common difference -2an = [? ]n + []
Given:
First term 9
Common difference -2
Required:
Find a formula for the nth term of the arithmetic sequence.
Explanation:
The general formula for the nth term of the an arithmetic sequence is given by the formula:
[tex]a_n=a+(n-1)d[/tex]Where a = first term
d = common difference
Put a = 9 and d = -2 in the formula.
[tex]\begin{gathered} a_n=9+(n-1)(-2) \\ a_n=9-2(n-1) \\ a_n=9-2n+2 \\ a_n=-2n+11 \end{gathered}[/tex]Final Answer:
The nth term of the arithmetic sequence is
[tex]a_n=-2n+11[/tex]Solve the system using substitution. You can eliminate the decimals if you like, but you don’t have to. The solution will be the same in either case. {0.45x + 0.10y = 4.30 y = 22-x (x,y)= (_, _)
SOLUTION:
Case: System of equations
Method:
[tex]\begin{gathered} 0.45x+0.10y=4.30....(1) \\ y=22-x....(2) \\ Substitute\text{ }y=22-x\text{ }into\text{ }eqn(1) \\ 0.45x+0.10y=4.30 \\ 0.45x+0.10(22-x)=4.30. \\ 0.45x+2.2-0.1x=4.30 \\ 0.45x-0.1x=4.3-2.2 \\ 0.35x=2.1 \\ x=\frac{2.1}{0.35} \\ x=\frac{210}{35} \\ x=6 \end{gathered}[/tex]Put x = 6 in y= 22 -x
[tex]\begin{gathered} y=22-x \\ y=22-6 \\ y=16 \end{gathered}[/tex]Final answer:
(x,y) = (6, 16)
help me with this simple math18. look for this simple math riddle1+4=52+5=123+6=218+11=?I'm stuck here when 8+11
Given,
The mathematical expressions are
1+4=5
2+5=12
3+6=21
8+11=?
The pattern of the expression is,
[tex]\begin{gathered} 1+1\times4=5 \\ 2+2\times5=12 \\ 3+3\times6=21 \end{gathered}[/tex]Similarly,
[tex]8+8\times11=96[/tex]Hence, the value is 96.
Calculate the area of each shape. Remember that the area of a triangle is: A= 1/2bh, and thearea of a rectangle is: A=bh1)2)E.dLaWG3)4)
The area of the (1)triangle is 1 square units and the area of the rectangle(5) is 24 square units,
Given area of a triangle is = 1/2 × b × h (b = base , h = height)
1) now from the figure we can see that the base of the triangle is 1 units and the height is 2 units,
∴area = 1/2 ×1 × 2 = 1 square units
2) now from the figure we can see that the base of the triangle is 2 units and the height is 4 units,
∴area = 1/2 ×2 × 4 = 4 square units
3) now from the figure we can see that the base of the triangle is 2 units and the height is 3 units,
∴area = 1/2 ×2 × 3 = 3 square units
4) now from the figure we can see that the base of the triangle is 3 units and the height is 2 units,
∴area = 1/2 ×3 × 2 = 3 square units
5) now from the figure we can see that the base of the rectangle is 6 units and the height is 4 units,
∴area of rectangle = b × h = 6 × 4 = 24 square units
6) now from the figure we can see that the base of the rectangle is 3 units and the height is 9 units,
∴area of rectangle = b × h = 3 × 9 = 27 square units
to learn more about area visit:
https://brainly.com/question/11952845
#SPJ9
i need help with math please
Information given
We know that Bryant ate 1/3 of a pizza
We also know that Brenda ate 1/2 of what Bryant ate
And we also know that Jack ate 1/2 a pizza more than Brenda
We want to find how much pizza did Jack eat
Notation
Let x the variable who represent a pizza
And for this case we can set upt the following equation:
[tex]x=B+Br+J[/tex]Where B = Bryant , Br= Brenda and J=Jack. We can replace the info given and we got:
[tex]x=\frac{1}{3}x+\frac{1}{2}(\frac{1}{3}x)+J[/tex]And solving for J we got:
[tex]J=x-\frac{1}{3}x-\frac{1}{6}x[/tex]We can take common facotr and we got:
[tex]J=x(1-\frac{1}{3}-\frac{1}{6})[/tex][tex]J=\frac{1}{2}x[/tex]Since Bryant ate 1/3 of the pizza we have remaining 1- 1/3= 2/3 of the pizza
From this 2/3 we know that brenda ate 1/6 so then we have remaining 2/3 -1/6= 1/2
So then Jack eat 1/6 + 1/12= 1/4 (Because Jack eat 1/2 a pizza more than Brenda)
I keep get 35 and it wrong? Can you please help me ?
Given: 7 different jellybeans
To Determine: How many ways the 7 different jellybeans can be lined up in a row of 3
Solution
We are considering an arrangement, so we would be using the permutation formula
The permuation formula is as shown below
[tex]_nP_r=\frac{n!}{(n-r)!}[/tex]Applying the formula above to the given as shown below
[tex]\begin{gathered} _7P_3=\frac{7!}{(7-3)!} \\ _7P_3=\frac{7\times6\times5\times4!}{4!} \\ _7P_3=7\times6\times5=210 \end{gathered}[/tex]Hence, the different ways 7 different jellybeans be lined up in a row of 3 is 210ways
let E be the event where the sun of two rolled dice is less than or equal to 4
If event E is defined as the event that "the sum of two rolled dice is less than or equal to 4."
Then the complement to E will represent all remaining outcomes, that is that "the sum of two rolled dice is greater than 4"
If the die is numbered from 1 to 6, the possible combinations are:
Add all combinations whose sums are greater than 4, without repeating combinations:
[tex]NºoutcomesE^c=3+4+4+3+2+1=17[/tex][tex]E^c=\mleft\lbrace(1,4);(1,5);(1,6);(2,3);(2,4);(2,5);(2,6);(3,3);(3,4);(3,5);(3,6);(4,4);(4,5);(4,6);(5,5);(5,6);(6,6)\mright\rbrace[/tex]-7(n - 2) + 2n = -5(n + 6)
Given the expression below
[tex]-7(n-2)+2n=-5(n+6)[/tex]To find n
Open the brackets
[tex]\begin{gathered} -7(n-2)+2n=-5(n+6) \\ -7n+14+2n=-5n-30 \\ -7n+2n+14=-5n-30 \\ -5n+14=-5n-30 \end{gathered}[/tex]Collect like terms
[tex]\begin{gathered} -5n+14=-5n-30 \\ -5n-(-5n)=-30-14 \\ 0\ne-44 \end{gathered}[/tex]Since, the sides are not equal,
Hence, there is no solution
A bag contains 6 red marbles, 10 green marbles, and 4 yellow marbles. You randomly pick a marble. What is the probability that it is a red or yellow? Write your answer as a reduced fraction (numerator/denominator).
Red marbles = 6
Green marbles = 10
Yellow marbles = 4
Total marbles in the bag = 20
The probability of picking a red or a yellow marble is;
[tex]\begin{gathered} Pr(\text{red or yellow)=Pr(picking red)}+Pr(\text{ picking yellow)} \\ Pr(\text{ picking red) = }\frac{n(red\text{ marbles)}}{total\text{ marbles}} \\ Pr(\text{ picking red) =}\frac{6}{20} \\ Pr(\text{ picking yellow)=}\frac{4}{20} \end{gathered}[/tex][tex]\begin{gathered} Pr(\text{ picking red or yellow)=}\frac{6}{20}+\frac{4}{20}=\frac{10}{20} \\ Pr(\text{ picking red or yellow)=}\frac{1}{2} \end{gathered}[/tex]The probability of picking a red or yellow marble is 1/2
Addison Stinson
Axis of Symmetry and Vertex (with Formula)
Nov 15, 8:00:24 PM
Find the equation of the axis of symmetry of the following parabola algebraically.
y = -2x² + 8x + 6
Answer:
?
Submit Answer
attempt 1 out of 2
how do you write 7,500,000,000,000,000,000 in scientific notation
7,500,000,000,000,000,000
In scientific notation, the 10 raised to a power having put the number to standard form
7,500,000,000,000,000,000
= 7.5 * 10^18
Brian likes to go bird watching along the Harvest Park Trail in a nearby forestpreserve. He wants to calculate the area enclosed by the trail which is shownbelow. Based on the diagram, what is the closest approximation of the area insquare yards of the land enclosed by the trail?
Answer:
(B)138,000 square yards.
Explanation:
To determine the area of the land enclosed by the trail, we divide the diagram into two as shown below:
Therefore, the area will be:
[tex]\begin{gathered} \text{Area}=(800\times900)+(400\times1300) \\ =720,000+520,000 \\ =1,240,000\text{ square feet} \end{gathered}[/tex]We then convert it to square yards.
[tex]\begin{gathered} \text{9 square feet=}1\text{ square yard} \\ \text{Therefore:} \\ 1,240,000\text{ square feet}=\frac{1,240,000}{9}\text{ square yard} \\ \approx137777\text{ square yards} \end{gathered}[/tex]Therefore, the closest approximation of the area in square yards is 138,000 square yards.
Solve the equation 3=4-5^3sqrtx^8
The solution of the equation is x =3.33.
What is a solution?A solution is a value assignment to an unknown variable that makes the equality of the equation true. In other words, a solution is a value or set of values (one for each unknown) that becomes an equation when the equation is replaced by the unknown. In mathematics, solving an equation means finding its solution, which is a value (number, function, set, etc.) that satisfies the conditions specified by the equation, usually two equations separated by an equal sign. Connected. When looking for a solution, one or more variables are called unknowns. A solution is a value assignment to an unknown variable that makes the equality of the equation true.In other words, a solution is a value or set of values (one for each unknown) that becomes an equation when the equation is replaced by the unknown.Solutions of equations are often called roots of equations and are not specifically limited to polynomial equations. The set of all solutions of an equation is the solution set.3 = 4 - 5³ × √x⁸
3 = 4 - 125 × x⁴
124 = x⁴
x = 3.33
To learn more about solution from the given link :
https://brainly.com/question/14603452
#SPJ1
The solution of the equation is x =3.33.
What is a solution?A solution is a value assigned to an unknown variable that makes the equality of the equation true.In other words, a solution is a value or set of values (one for each unknown) that becomes an equation when the equation is replaced by the unknown.In mathematics, solving an equation means finding its solution, which is a value (number, function, set, etc.) that satisfies the conditions specified by the equation, usually two equations separated by an equal sign.A solution is a value assigned to an unknown variable that makes the equality of the equation true.Solutions of equations are often called roots of equations and are not specifically limited to polynomial equations.The set of all solutions of an equation is the solution set:
3 = 4 - 5³ × √x⁸3 = 4 - 125 × x⁴124 = x⁴x = 3.33Therefore, the solution of the equation is x =3.33.
To learn more about the solution from the given link :
brainly.com/question/14603452
#SPJ1
i need help with this HW problem it is ,the length of a rectangle is 2 more thn 3 times the width.If the perimeter is 46 ,si what would the length and width be?
If w is the width of the rectangle, then, you have for the length l of the rectangle:
l = 3w + 2
take into account that the perimeter of the rectangle is 46, and the expresionf for the perimeter P is:
P = 2l + 2w
in order to determine the value of the length l, replace the expression
l = 3w + 2 into the expression for the perimeter P, then, solve for w:
P = 2(3w + 2) + 2w
P = 6w + 4 + 2w
P = 8w + 4
replace P = 46:
46 = 8w + 4
46 - 4 = 8w
42 = 8w
42/8 = w
21/4 = w
5.25 = w
replace the previous value of w into the expression l = 3w + 2
l = 3(5.25) + 2
l = 17.75
Hence, the length of the rectangle is 17.75
O.
O
45
39
IS
15
6 2
If Alm and mz6= 4x - 15 and m₂7 = x + 30, then mz6=
35
3
8
7
m
In the given figure, the measure of angle ∠6 is (B) 45°.
What are angles?When two straight lines or rays intersect at a single endpoint, an angle is created.
The vertex of an angle is the location where two points come together.
The Latin word "angulus," which means "corner," is where the term "angle" originates.
So, the measure of angle 6:
We know that
∠6 = 4x - 15
∠7 = x + 30
We can easily tell while looking at the figure that ∠7 and ∠6 are alternate angles which means they are equal.
So, we can write this is:
4x - 15 = x + 30
3x = 45
x = 15
Now, substitute x = 15 in 4x - 15:
4x - 15
60 - 15
45
Therefore, in the given figure, the measure of angle ∠6 is (B) 45°.
Know more about angles here:
https://brainly.com/question/29276964
#SPJ1
The volume of a cube is 27000 cubic inches. What is the length of one side?
Given:
Volume of a cube = 27,000 in^3
(Note: A cube has equal sides)
The volume of a cube = a^3
So,
[tex]\begin{gathered} 27000=a^3 \\ \sqrt[3]{27000}\text{ = a} \\ a\text{ = 30 in.} \end{gathered}[/tex]Therefore, the lenght of one side is 30 inches.
Factor this question Q3+125
write 125 as a power in base 5
[tex]q^3+5^3[/tex]then, apply the rule for the addition of cubes
[tex]\begin{gathered} a^3+b^3=(a+b)\cdot(a^2-a\cdot b+b^2) \\ \text{then, } \\ q^3+5^3=(q+5)\cdot(q^2-5x+5^2) \\ q^3+5^3=(q+5)\cdot(q^2-5x+25) \end{gathered}[/tex]2. Kiran is spending $12 on games and rides at another carnival, where a game costs $0.25 and a ride costs $1. Which graph represents therelationship between the quantities in this situation? Explain how you know.
Explanation: The graph represents the relation between how many dollars were spent on rides and games at the same time once Kiran has just $12 to spend.
First: Once Kiran has just $12 to spend we know the maximum must be 12 rides once 1 ride costs $1.
Second: Let's imagine Kiran spends all the $12 dollars in rides, it means there is nothing left for the games which means she spends $0 on games (number of games = 0).
Third: Now let's imagine Kiran spent just $10 on rides ($10 on rides = 10 rides), so she has still $2 to spend on games. Once each game costs $0.25 it means Kiran is able to pay for 8 games (2/0.25=8) this time.
Fourth: Now let's imagine Kiran spent just $8 on rides ($8 on rides = 8 rides), so she has still $4 to spend on games. Once each game costs $0.25 it means Kiran is able to pay for 16 games (4/0.25=16) this time.
Final answer: As we can see the only graph that represents the pattern represented above is the letter C.
Question 11 of 25 Which of the following functions is graphed below? A. y = x +51 + 4 B. V = x + 51-4 C. y = x-5|+4 D. y = x-51-4
The simple way to answer this is to take a few points from the graph and compare it with each option.
Let's take points at x = 0, 1, 2
From graph,
When x = 0, y = 1, hence the point is (0, 1)
When x = 1, y = 0, hence the point is (1, 0)
When x = 2, y = -1, hence the point is (2, -1)
Now, put the same x values in the given options to evaluate the output.
For x = 0, the pair should be (0, 1),
1) |x + 5| + 4 = |0 + 5| + 4 = |5| + 4 = 9 => (0, 9) Not true
2) |x + 5| - 4 = |0 + 5| - 4 = |5| - 4 = 1 => (0, 1) True
3) |x - 5| + 4 = |0 - 5| + 4 = |-5| + 4 = 5 + 4 = 9 => (0, 9) Not True
4) |x - 5| - 4 = |0 - 5| - 4 = |-5| - 4 = 5 - 4 = 1 => (0, 1) True
Hence, 2nd and 4th options can be true. Now, evaluate these two options with some other point.
For x = 1, the pair should be (1, 0),
2) |x + 5| - 4 = |1 + 5| - 4 = |6| - 4 = 2 => (0, 2) Not True
4) |x - 5| - 4 = |1 - 5| - 4 = |-4| - 4 = 4 - 4 = 1 => (0, 0) True
Hence, the 4th option is true.
solve the system x+3y=62x+4y=12
x+3y=6 ----------------------------(1)
2x+4y=12---------------------------(2)
Using elimination method to solve;
we will eliminate x variable
To do that, we must make sure the coefficient of x in the two equation are the same.
This can be achieved by multiplying equation (1) by 2 and equation (2) by 1
That is;
2x + 6y = 12 ----------------------(3)
2x + 4y = 12 -----------------------(4)
subtract equation (4) from equation (3)
2y = 0
Divide both-side of the equation by 2
y=0
substitute y = 0 into equation (1)
x + 3(0) = 6
x = 6
Two mechanics worked on a car. The first mechanic charged 65 per hour and the second Mechanic charged 100 per hour. The mechanics worked for a combined total of 25 hours and together they charged a total of $2150. How long did each mechanic work
To solve this problem we have to write an equation for each condition where the first charge is x and the second charge is y so:
For the total hours will be:
[tex]x+y=25[/tex]and the total charge will be:
[tex]65x+100y=2150[/tex]We can solve the first equation for x so:
[tex]x=25-y[/tex]and we replace that in the secon equation so:
[tex]65(25-y)+100y=2150[/tex]and we solve for y so:
[tex]\begin{gathered} 1625-65+100y=2150 \\ 35y=2150-1625 \\ y=\frac{525}{35} \\ y=15 \end{gathered}[/tex]And with this value of y we can find x so:
[tex]\begin{gathered} x=25-15 \\ x=10 \end{gathered}[/tex]If beginning finished good rupees 45000 and cost of goods manufactured rupees 25,000 and ending finished goods to go to rupees 50,000 then what is the value of cost of goods sold:Answer 20000
Answer:
20000
Step-by-step explanation:
Answer:
Hello, just a slight bit confused by your phrasing?
I would love to help, I just can't seem to understand how.
Step-by-step explanation:
Just let me know, here to help!
how do u find weather the system has one solution,no solutiin,solution,infinitely many solutions
1st case: the system has one solution
For example:
Given this system:
[tex]\begin{gathered} x+y=5 \\ 2x-y=4 \\ \text{Let:} \\ x+y=5\text{ (1)} \\ 2x-y=4\text{ (2)} \\ \text{ Using elimination method:} \\ (1)+(2)\colon \\ x+2x+y-y=9 \\ 3x=9 \\ x=\frac{9}{3} \\ x=3 \\ y=5-x \\ y=5-3 \\ y=2 \end{gathered}[/tex]graphically, a system has a solution if the two lines intersect, the point of intersection is the solution.
--------------------------------------------
2nd case: the system has no solution
A system has no solution, when the lines are parallel and have different intercepts, for example:
[tex]\begin{gathered} y=2x+1 \\ y=2x-3 \end{gathered}[/tex]as you can see the lines never cross each other.
3rd case: the system has infinitely many solutions
occurs when one line is a scalar multiple of the other, in other words it is the same line. for example:
[tex]\begin{gathered} x+y=5 \\ 2x+2y=10 \end{gathered}[/tex]