Determine the value of angle theta.
[tex]\begin{gathered} \sec \theta=\frac{9}{4} \\ \theta=\sec ^{-1}(\frac{9}{4}) \\ =296.3878\text{ (As }\theta\text{ lie in fourth quadrant)} \end{gathered}[/tex]Determine the value of tan theta.
[tex]undefined[/tex]Line AB is perpendicular to line AC, line CD is congruent to line CE and measurement of angle B is 48° find measurement of angle DEB
The sum of the interior angles of a triangle adds up to 180°.
Based on this, we can do the following...
0. Finding m∠C:
[tex]m\angle C+m\angle B+m\angle A=180[/tex][tex]m\angle C=180-m\angle B-m\angle A[/tex]With the description of the problem, we know that m∠B = 48° and m∠A = 90°. Replacing these values:
[tex]m\angle C=180-48-90[/tex][tex]m\angle C=42[/tex]With this angle and based on the same logic that the addition of the interior angles of a circle adds up 180°, we can get m∠CED. Also, as CD is congruent to CE, m∠CED = m∠CDE.
[tex]m\angle C+m\angle CED+m\angle CDE=180[/tex][tex]m\angle C+m\angle CED+m\angle CED=180[/tex][tex]m\angle C+2m\angle CED=180[/tex][tex]m\angle CED=\frac{(180-m\angle C)}{2}[/tex]Replacing the value of m∠C previously calculated:
[tex]m\angle CED=\frac{(180-42)}{2}=\frac{138}{2}[/tex][tex]m\angle CED=69[/tex]Finally, as we know segment CB is a straight line, the angle is 180°. Thus...
[tex]m\angle DEB+m\angle CED=180[/tex][tex]m\angle DEB=180-m\angle CED[/tex]Replacing the value previously calculated:
[tex]m\angle DEB=180-69[/tex]Answer:
[tex]m\angle DEB=111[/tex]The data points show the amount of money y (in dollars) in an account after a time x (in years).Each figure has the same data points.However, each figure has a different curve fitting the data.The equation for each curve is also shown.Answer the questions that follow.
SOLUTION
(a) From the slope of the graph, we can see that this is an exponential function
So, the function that best fits the graph is
[tex]y=601(1.05)^x[/tex]Hence the answer to (a) is Figure 1
(b) From the graph of the function we selected, where y represents the amount of money and x the number of years. To get the amount of money, we will substitute x for 18 into the quatiomn, we have
[tex]\begin{gathered} y=601(1.05)^x \\ y=601(1.05)^{18} \\ y=601\times2.406619234 \\ y=1,446.378159448 \end{gathered}[/tex]Hence the answer is $1,446.38
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)
80 plus 8 plus 76 plus 5 plus 50 plus 75 plus 100 plus 100
80 plus 8 plus 76 plus 5 plus 50 plus 75 plus 100 plus 100 in Mathematical equation will be:
[tex]80\text{ + 8 + 76 + 5 + 50 + 75 + 100 +100}[/tex]Let's determine the sum,
[tex]80\text{ + 8 + 76 + 5 + 50 + 75 + 100 +100}[/tex][tex]88\text{ + 76 + 5 + 50 + 75 + 2}00[/tex][tex]164\text{ + 5 + 50 + 2}75[/tex][tex]169\text{ + 3}25[/tex][tex]494[/tex]Therefore, the answer is 494.
Answer:
496
Step-by-step explanation:
80+8+76+5+50+75+100+100
First, Narrow it down a bit: (80+8)=88, (76+5)=81, (50+75)=125, and (100+100)=200.
Now, your equation is: 88+81+125+200.
Now, narrow it down a bit more: (88+81)=169 and (125+200)=325.
Now, your equation is just: 169+325
Lastly, narrow it down to the last number: (169+325)=496.
Define an exponential function, h(x), which passes through the points (1,16) and(5, 1296). Enter your answer in the form axb^xh(x) =
Define an exponential function, h(x), which passes through the points (1,16) and
(5, 1296). Enter your answer in the form axb^x
the equation is of the form
[tex]y=a(b)^x[/tex]we have
point (1,16)
so
For x=1, y=16
substitute
[tex]\begin{gathered} 16=a(b)^1 \\ 16=a\cdot b \end{gathered}[/tex]isolate the variable a
[tex]a=\frac{16}{b}[/tex]Point (5,1296)
For x=5, y=1,296
substitute
[tex]1,296=a(b)^5[/tex]substitute equation 1 in equation 2
[tex]1,296=(\frac{16}{b})\cdot b^5[/tex]solve for b
[tex]\begin{gathered} \frac{1296}{16}=b^4 \\ b^4=81 \\ b=3 \end{gathered}[/tex]Find the value of a
a=16/3
therefore
the equation is
[tex]y=\frac{16}{3}\cdot(3)^x[/tex]see the attached figure to better understand the problem
Walk me through step by step for the third question
The First Question:
√8
Prime factorization of 8: 2³
= √2³
Apply radical rule: a^b+c = a^b * a^c
2³ = 2² * 2
=√2² * 2
Apply radical rule: √ab = √a√b, a ≥ 0, b ≥ 0
√2² * 2 = √2² √2
Apply radical rule: √a² = a, a ≥ 0
√2² = 2
2√2
The Third question:
√-8
Let's apply radical rule:
√-a = √-1 √
How many units the function y = |x + 7| is translated from the parent function?Moved 7 units left from the originMoved 7 units right from the originMoved 7 units upwards from the originMoved 7 units downwards from the origin
According to the given information, the correct answer is the first choice Moved 7 units left from the origin.
The parent function would be |x|, and if we have |x+7| all the function values will be moved 7 units left from the origin.
I need help. I need to know the measure of
The figure appears to be a rectangle and its opposite sides are parallel.
With this condition, ∠FDB and ∠EBG should be alternating angles same with ∠BED and ∠DFE. Under the rules of alternate angles, the two angles should be equal.
Therefore, ∠FDB = ∠EBG and ∠FDB = 27°
∠EBG should also be equal to 27°
The answer is 27°
Given the set of data below, which measure(s) will change if the outlier is removed? (Check all that apply.)1, 6, 8, 8, 8meanrangemedianmode
Answer
mean
range
Step-by-step explanation
An outlier is an observation that lies an abnormal distance from the other values.
In the set of data:
[tex]1,6,8,8,8[/tex]the outlier is 1.
Given that the mean is the average between all values in the dataset, then if the outlier is removed, the mean changes.
The range is calculated as follows:
[tex]range=maximum-minimum[/tex]If 1 is removed, the minimum changes, and in consequence, the range also changes.
The median is the middle number in a sequence of numbers. In this case, we have:
The median is 8 in both cases.
The mode is the value that appears most often in a set of data values. The mode of the original dataset is 8, and if the outlier is removed, the median remains the same.
x²+25factor and find gcf first
First, write out the binomial expression.
[tex]x^2\text{ + 25}[/tex]The binomial expression is a prime mathematical express meaning it has two factors, 1 and x^2 + 25.
Therefore,
[tex]\begin{gathered} \text{The factors of x}^2+25 \\ \text{are 1 and (x}^2\text{ + 25).} \end{gathered}[/tex]If LW = 5x + 2 and LJ = 11x + 2 in the parallelogram below. Find LW.
Recall that in a parallelogram, the diagonals bisect each other. That means they divide each other in exactly equal parts.
that means that if LW = 5 x + 2 , and LJ = 11 x + 2,
since LJ is the full diagonal segment, and LW is half of it, we can say:
LJ = 2 times LW
In mathematical terms:
LJ = 2 * (LW)
11 x + 2 = 2 * (5 x + 2)
use distributive property
11 x + 2 = 10 x + 4
subtract 10 x from both sides
11 x - 10 x + 2 = 4
x + 2 = 4
subtract 2 from both sides to isolate x completely on the left
x = 4 - 2
x = 2
Then now that we know the value of x, we can use it in the formula for LW:
LW = 5 x + 2 = 5 * 2 + 2 = 10 + 2 = 12
Then LW = 12
Then answer option A is the correct one.
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]Cooper needs to order some new supplies for the restaurant where he works. The restaurant needs at least 555 spoons. There are currently 191 spoons. If each set on sale contains 10 spoons, clearly Write and solve an inequality which can be used to determine x, the number of sets of spoons Cooper could buy for the restaurant to have enough spoons.
x ≥ 36.4
Explanation:At least 555 means it can be equal to or greater than 555.
It is represented as ≥ 555
Amount of spoons available = 191
let the number of set = x
Since there are 10 spoon in each set, total set we can have:
10 × x = 10x
The equation becomes:
Total number of sets + Amount of spoons available ≥ 555
10x + 191 ≥ 555
Solving the equation:
10x + 191 ≥ 555
collect like terms by subtracting 191 from both sides:
10x + 191 - 191 ≥ 555 - 191
10x ≥ 364
divide both sides by 10:
10x/10 ≥ 364/10
x ≥ 36.4
The number of sets that Cooper could buy would be equal to 36.4 or greater than 36.4
Find an equation for the ellipse whose vertices are at (4,-3) and (4,7), and focus is at (4,4).
The vertices of the ellipse are given as
[tex]\begin{gathered} V_2=(4,-3) \\ V_1=(4,7) \end{gathered}[/tex]The focus of the ellipse is given as
[tex](4,4)[/tex]The equation of an ellipse is given as
[tex]\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1[/tex]Where the coordinate of the center is
[tex](h,k)[/tex]To calculate the coordinate of the center, we will use the formula below
[tex]\begin{gathered} \frac{(x_1+x_2)}{2},\frac{(y_1+y_2)}{2} \\ =\frac{(4+4)}{2},\frac{(-3+7)}{2} \\ =\frac{8}{2},\frac{4}{2} \\ (4,2) \\ (h,k)=(4,2) \end{gathered}[/tex]The formula to calculate the value of a is given below
[tex]\begin{gathered} V_1=(h,k+a) \\ V_2=(h,k-a) \end{gathered}[/tex]By comparing coefficients, we will have
[tex]\begin{gathered} k+a=7\ldots\text{.}(1) \\ k-a=-3\ldots\text{.}(2) \end{gathered}[/tex]By adding equations (1) and (2) and solving simultaneously, we will have
[tex]\begin{gathered} 2k=4 \\ \text{divide both sides by 2,} \\ \frac{2k}{2}=\frac{4}{2} \\ k=2 \end{gathered}[/tex]By substituting hk=2 in equation 1, we will have
[tex]\begin{gathered} k+a=7 \\ 2+a=7 \\ a=7-2 \\ a=5 \end{gathered}[/tex]The coordinate of the focus,is calculated using the formula below
[tex](h,k+c)[/tex]By substituting the values, we will have
[tex]\begin{gathered} k+c=4 \\ 2+c=4 \\ c=4-2 \\ c=2 \end{gathered}[/tex]The value of will be calculated using the formula below
[tex]\begin{gathered} c^2=a^2-b^2 \\ b^2=a^2-c^2 \end{gathered}[/tex]By substituting the values, we will have
[tex]\begin{gathered} b^2=a^2-c^2 \\ b^2=(5)^2-(2^2 \\ b^2=25-4 \\ b^2=21 \end{gathered}[/tex]By substituting the values of a,b,h and k in the equation of an ellipse, we will have
[tex]\begin{gathered} \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 \\ \frac{(x-4)^2}{21^{}}+\frac{(y-2)^2}{25^{}}=1 \\ \end{gathered}[/tex]Hence,
The equation of the ellipse will be
[tex]\frac{(x-4)^2}{21^{}}+\frac{(y-2)^2}{25^{}}=1[/tex]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]Work out the value of (2 cubed) squared
Answer: 64
Step-by-step explanation:
2 cubed is 8 8 squared is 64 and that is the answer
Solve the system of equations using elimination.15q – 4r = 625q + 8r = 86A.q = –7, r = –6B.q = 6, r = 7C.q = –7, r = –7
To solve the system we multiply the first equation by 2, then we get:
[tex]\begin{gathered} 30q-8r=124 \\ 5q+8t=86 \end{gathered}[/tex]Now we add the equations:
[tex]\begin{gathered} 35q=210 \\ q=\frac{210}{35} \\ q=6 \end{gathered}[/tex]Once we know the value of q we plug it in the first equation and solve for r:
[tex]\begin{gathered} 15(6)-4r=62 \\ 90-4r=62 \\ 28=4r \\ r=\frac{28}{4} \\ r=7 \end{gathered}[/tex]Then q=6 and r=7 and the answer is B.
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
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.
Multiply 3/4 × 16/9 O A. 2/4O B. 3/4O C. 64/27O D. 4/3
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given question
[tex]\frac{3}{4}\times\frac{16}{9}[/tex]STEP 2: Find the product
[tex]\begin{gathered} \frac{3}{4}\times\frac{16}{9} \\ Muliply\text{ the numerators and also the denominators} \\ =\frac{3\times16}{4\times9}=\frac{48}{36} \\ \\ Divide\text{ by the common factors} \\ \frac{48}{36}=\frac{12\times4}{12\times3}=\frac{4}{3} \end{gathered}[/tex]Hence, the result is 4/3
Find the measure of the indicated angle in each triangle. These are the answer A=360 B=129 C=80 D=180
1) The best way to tackle this question is to remind ourselves of the Exterior Angle Theorem.
2) The exterior angle theorem states that an exterior angle is equal to the sum of two remote angles within the triangle.
So let's find m∠QRX:
[tex]\begin{gathered} m\angle QRX=49^{\circ}+80^{\circ} \\ m\angle QRX=129^{\circ} \end{gathered}[/tex]Megan and her friends went to the movies. Megan took $45 with her to spend on her favorite items so she could share with her friends. Megan loves both popcorn and candy. The price for each bag of popcorn was $9. The price of each box of candy is half the price of a bag of popcorn. a). Sketch the graph that represents the situation and label the intercepts. Use one axis to represent the number of bags of popcorn and the other axis to represent the number boxes of candy.b). Explain your graph.
Solution:
Given that;
Megan and her friends went to the movies.
Megan took $45 with her to spend on her favorite items so she could share with her friends. Megan loves both popcorn and candy
Let x represent the number of bags of popcorn bought
Let y represent the number of boxes of candy bought
The price for each bag of popcorn was $9. The price of each box of candy is half the price of a bag of popcorn, i.e.
The price of a box of candy will be
[tex]=\frac{9}{2}=\text{\$}4.5[/tex]The equation representing the situation is
[tex]9x+4.5y=45[/tex]a) The graph representing the situation is
The intercepts are
[tex]\begin{gathered} x-intercept:\text{ \lparen5,0\rparen} \\ y-intercept:\text{ }(0,10) \end{gathered}[/tex]b) From the graph:
The graph explains that;
Megan bought 5 bags of popcorn and 10 boxes of candy
00What is the length of BC?17А A9 units11 unitsO 15 unitsO 16 units8С
ANSWER:
15 units
STEP-BY-STEP EXPLANATION:
We can determine the BC value we can determine it by means of the Pythagorean theorem, with the hypotenuse 17 and the other side 8, therefore:
[tex]\begin{gathered} h^2=a^2+b^2 \\ \\ \text{ We replacing} \\ \\ 17^2=8^2+b^2 \\ \\ b^2=289-64 \\ \\ b=\sqrt{225} \\ \\ b=15 \end{gathered}[/tex]Therefore, the length of BC is 15 units.
-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
the set of ordered pairs below represent a linear equation (-2,-3),(0,-2),(2,-1),(x,y)what is one other pair of coordinates that could be in the missing ordered pair,(x,y), in this set?
Let's calculate the relationship between the given points
We can see that an increase of +2 units on the x-axis corresponds to an increase of +1 unit on the y-axis.
With this in mind, we can say that an additional point would be to
(4, 0)
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
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
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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 :
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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
witch conversion factor would you use to convert from meters to feel ?
The conversion factor = 0.3048
Explanations:The conversion from meters to feet is given by the formula:
1 foot = 0.3048 meters
Therfore, to convert a measurement from meters to feet, it has to be multiplied by 0.3048.
The conversion factor = 0.3048
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]