Which statements are true about the graph of y s 3x + 1 and y 2 -x + 2? Check all that apply.
O The slope of one boundary line is 2.
O Both boundary lines are solid.
A solution to the system is (1, 3).
O Both inequalities are shaded below the boundary lines
O The boundary lines intersect.
Answers
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Explanation:
Let's go through each of the answer choices to see which are true and which are false.
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Choice A
The general slope intercept form is y = mx+b
m = slope and b = y intercept.
For y = 3x+1, the boundary line of the first inequality, the slope is 3.
For y = -x+2, the slope is -1.
None of the slopes are 2.
Choice A is false.
We'll use the concept of slope again in choice E.
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Choice B
The "or equal to" part of the inequality sign is what directly determines the boundary line being solid. This is because we include points on the boundary.
Choice B is true.
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Choice C
Plug x = 1 and y = 3 into the first inequality
[tex]y \le 3x+1\\\\3 \le 3(1)+1\\\\3 \le 3+1\\\\3 \le 4\\\\[/tex]
The last inequality is true, so the first inequality is true when (x,y) = (1,3). This makes (1,3) a solution to [tex]y \le 3x+1\\\\[/tex]
Repeat those steps for the other inequality given to us
[tex]y \ge -x+2\\\\3 \ge -1+2\\\\3 \ge 1\\\\[/tex]
which is also true, so that makes (1,3) also a solution to [tex]y \ge -x+2\\\\[/tex]
The point (1,3) is a solution to both inequalities at the same time, making it a solution to the system overall.
Choice C is true
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Choice D
When y is fully isolated, the "less than" inequality indicates the shading is below the boundary. This is due to us considering points of y coordinates smaller than the boundary line. So we'll shade below the boundary for [tex]y \le 3x+1\\\\[/tex]
We don't do the same for [tex]y \ge -x+2\\\\[/tex]. Instead, we'll shade above the solid boundary line.
Choice D is false.
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Choice E
If two lines have equal slopes, but different y intercepts, then we have parallel lines. Parallel lines never intersect.
If on the other hand, the slopes are different values, then the two lines will intersect somewhere on the xy grid. There will only be exactly one point of intersection.
The given system has boundary lines of slopes 3 and -1, for the first and second inequality respectively. Refer back to choice A. These differing slopes tell us that the boundary lines intersect somewhere.
Choice E is true
Let's go through all the options one at a time and look for the correct ones
Option 1: The slope of one boundary line is 2
We have 2 equations of lines, where the coefficients of are 3 and -1 respectively
because the coefficient of x denotes the slope of a line, we know that the lines have the slope 3 and -1, not 2
Hence, this option is Incorrect
Option 2: Both boundary lines are solid
In order for the boundary lines to be solid, the inequality must have an 'equal to', like ≤ (less than or equal to) or ≥ (greater than or equal to)
we can see that that's the case in our case and hence, this option is Correct
Option 3: A solution to the system is (1, 3)
To confirm this, we'll plug these coordinates into the given inequalities and see if it stands correct
y ≤ 3x + 1
3 ≤ 3(1) + 1
3 ≤ 4 which is correct because 3 is less than 4
Second equation:
y ≥ 2 - x
3 ≥ 2 - 1
3 ≥ 1
Which is also true because 3 is greater than 1
Now, we can say that (1 , 3) is a solution to the system because it satisfies both the equations and is Correct
Option 4: Both inequalities are shaded below the boundary lines
For an inequality to be shaded below the boundary line, it must have the ≤ inequality (in case of solid line) and < inequality (in case of dotted line)
because the second inequality listed includes the ≥ inequality, which was not mentioned above, it won't be shaded below
another way to think about it is that any 'greater than' inequality will shade everything above the line and the 'lesser than' inequality will shade below the line
which means that this option is Incorrect
Option 5: The boundary lines intersect
In order for the boundary lines to intersect, they must have have different slopes.
as we mentioned in the explanation of the first option, that the slopes of the lines is 3 and -1, which are different slopes
Therefore, this option is Correct
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Answers
Answer:
P
[tex] \alpha [/tex]
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P=k/Q²
K=P×Q²
Where P=9
Q=8
K=?
K=9×8²
K=9×64
K=573
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Use the Pythagorean theorem. Name the hypotenuse (longest side) c and the bottom side b. Then, simplify and solve for a.
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9514 1404 393
Answer:
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Step-by-step explanation:
Solving for t, we find ...
A = Pe^(rt)
A/P = e^(rt)
ln(A/P) = rt
t = ln(A/P)/r
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Answers
Answer:
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Step-by-step explanation:
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Step-by-step explanation:
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Answers
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Answer:
12 $5 bills
Step-by-step explanation:
Represent the number of each kind of bill by x, y and z: there are x $1 bills, y $5 bills and z $10 bills.
According to the problem statement,
x + y + z = 25 and y = 4z (or z = y/4). Also, the values of the bills add up to $100:
($1)x + ($5)y + ($10)z = $100. Eliminate z by typing y/4 in its place:
x + y + (y/4) = 25 and
x + 5y + 10(y/4) = 100
Doing this has reduced the number of variables to two: x and y. Combining like terms in both equations, we get:
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Answers
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Answers
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Answers
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Answers
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Answer:
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-----------------
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Answers
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Answers
Answer:
100x + 13000y = 865000
Step-by-step explanation:
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Answers
Step-by-step explanation:
If you divide 1,000 by 2 since there are £2, you get 500. Multiply 500 by 25 and you got it!
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What is the location of the point on the number line that is 3/5 of the way from
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O A. 12
O B. 10
C. 9
O D. 11
Answers
The location of the point on the number line that is 3/5 of the way from A = 2 to B = 17 is 11
How to determine the location of the point?The points are given as:
A = 2
B = 17
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The location of the point is then calculated as:
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So, we have:
Point = 2 + 3/5(17 - 2)
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Answers
Answer:
6 inches
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the formula for the volume of a rectangular prism is width*length*height.
Here we are asked to find the height when given a volume. The resultant equation is:
volume = w * l * h
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192 cu.in./32 sq.in.= h inches
6 inches = height
Translations, rotations, and reflections are rigid transformations. What can you conclude about the measures of sides and angles on any triangle after undergoing a series of rigid transformations? Explain.
Answers
Answer:
After a translation, the measures of the sides and angles on any triangle would be the same since translation only involves changing the coordinates of the vertices of the triangle.
After a rotation, the measures of the sides and angles of a triangle would also be the same. Similar to translation, the proportion of the triangle is unchanged after a rotation.
After a reflection, the triangle's sides and angles would still be the same since reflection is a rigid transformation and the proportion of the sides and angles are not changed.
Step-by-step explanation:
Rigid transformations, i.e. translations, rotations, and reflections, preserve the side lengths and angles of any figure. Therefore, after undergoing a series of rigid transformations, the side lengths and angle measures of any triangle will be the same as the original triangle, generally speaking, in another position.
Answer:
They are all congruent or equal. The angles and side lengths are preserved through the rigid transformations.
Step-by-step explanation:
That’s what i put