Find the coordinates of the intersection point that solves the system of these two linear equations in two variables: 2x + y = 15 and y = −x + 5.
On a graph there's 3x - 2y = -4 and x - 5y = 3.
A. (10, −5)
B. (5, −10)
C. (−10, 5)
D. (−5, 10)
To find the intersection point of the two equations 2x + y = 15 and y = -x + 5, you can solve them simultaneously.
Firstly, substitute y = -x + 5 into the first equation:
2x + (-x + 5) = 15
2x - x + 5 = 15
x + 5 = 15
x = 10
Next, substitute x = 10 back into the second equation to find y:
y = -10 + 5
y = -5
Therefore, the intersection point is (10, -5).
For the second set of equations 3x - 2y = -4 and x - 5y = 3, you can use the same method to solve for the intersection point.
Firstly, multiply the second equation by 3:
3(x - 5y) = 3(3)
3x - 15y = 9
Now, rewrite the equations as a system:
3x - 2y = -4
3x - 15y = 9
Subtract the first equation from the second equation:
-13y = 13
y = -1
Substitute y = -1 back into the first equation to solve for x:
3x - 2(-1) = -4
3x + 2 = -4
3x = -6
x = -2
Therefore, the intersection point for the second set of equations is (-2, -1).
The correct answer is not listed among the options given.
Oh I'm sorry the equations on the chart are y = -x + 5, 2x + y = 15
No problem! Let's solve the system of equations y = -x + 5 and 2x + y = 15.
Substitute y = -x + 5 into the second equation:
2x + (-x + 5) = 15
2x - x + 5 = 15
x + 5 = 15
x = 10
Now, substitute x = 10 back into the first equation to find y:
y = -10 + 5
y = -5
Therefore, the intersection point of the equations y = -x + 5 and 2x + y = 15 is (10, -5).
The given options are:
A. (10, −5)
B. (5, −10)
C. (−10, 5)
D. (−5, 10)
So, the correct answer based on the intersection point we found is A. (10, -5).
Is (−2, 6) a solution to the system of these linear equations: x + 2y = 10 and 3x + y = 0? Why?
Responses:
No, because the graphs don’t intersect at (−2, 6).
Yes, because the graphs intersect at (−2, 6).
Yes, because the graphs don’t intersect at (−2, 6).
No, because the graphs intersect at (−2, 6).
To determine if (-2, 6) is a solution to the system of equations x + 2y = 10 and 3x + y = 0, you need to substitute x = -2 and y = 6 into both equations to see if they satisfy the equations.
1. Substitute x = -2 and y = 6 into x + 2y = 10:
-2 + 2(6) = 10
-2 + 12 = 10
10 = 10
The equation is true.
2. Substitute x = -2 and y = 6 into 3x + y = 0:
3(-2) + 6 = 0
-6 + 6 = 0
0 = 0
The equation is true.
Since (-2, 6) satisfies both equations, it is a solution to the system.
Therefore, the correct response is:
Yes, because the graphs intersect at (-2, 6).
Estimate the solution to the system of these linear equations based on the graph.
y=5x+10
y=−2x+3
On a graph there's y = -2x + 3, y = 5x + 10.
Responses
A. (−1, 5)
B. (−1,−5)
C. (1, 5)
D. (5,−1)
Based on the equations y = -2x + 3 and y = 5x + 10, you can estimate the solution to the system of linear equations by identifying the point where the two lines intersect on the graph.
The correct answer depends on where the two lines intersect. Let's analyze the slopes of the two lines:
1. The line y = -2x + 3 has a negative slope (-2) and a y-intercept of 3.
2. The line y = 5x + 10 has a positive slope (5) and a y-intercept of 10.
By observing the equations and the slopes, the lines y = -2x + 3 and y = 5x + 10 intersect at approximately (-1, 5) on the graph.
Therefore, the best estimate based on the graph is:
C. (1, 5)
What is the last step to solving a system of equations?(1 point)
Responses:
Graph the lines.
Estimate the intersection point.
Make a table of solutions of the linear equations.
Check the answer.
The last step to solving a system of equations is typically to check the answer.
Therefore, the correct response is:
Check the answer.
Estimate the solution to the system of these linear equations based on the graph.
y=−5x+7
y=−2x+10
Responses:
A. (−1,−12)
B. (−1, 12)
C. (12,−1)
D. (1,−12)