Solving Systems of Equations by Graphing Quick Check
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Question
Use the image to answer the question.
A coordinate plane with four quadrants shows the x-axis ranging from negative 5 to 15 in increments of 1 and the y-axis ranging from negative 10 to 10 in increments of 1. A solid line and a dotted line intersect each other. The equation of the solid line is 2 x plus y equals 15. The equation of the dotted line is y equals negative x plus 5. The intersection of both lines is shown at positive 10 on the x-axis and negative 5 on the y-axis in quadrant 4.
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 .
(1 point)
Responses
(5,−10)
left parenthesis 5 comma negative 10 right parenthesis
(−10, 5)
left parenthesis negative 10 comma 5 right parenthesis
(−5, 10)
left parenthesis negative 5 comma 10 right parenthesis
(10,−5)
left parenthesis 10 comma negative 5 right parenthesis
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(10, -5)
Is (−2, 6) a solution to the system of these linear equations: x+2y=10 and 3x+y=0 ? Why?(1 point)
Responses
No, because the graphs don’t intersect at (−2, 6).
No, because the graphs don’t intersect at left parenthesis negative 2 comma 6 right parenthesis .
Yes, because the graphs intersect at (−2, 6).
Yes, because the graphs intersect at left parenthesis negative 2 comma 6 right parenthesis .
No, because the graphs intersect at (−2, 6).
No, because the graphs intersect at left parenthesis negative 2 comma 6 right parenthesis .
Yes, because the graphs don’t intersect at (−2, 6).
No, because the graphs don't intersect at (-2, 6).
Use the image to answer the question.
An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. A solid line and a dotted line with arrows at both the ends intersect each other. The equation of the solid line is y equals 5 x plus 10. The equation of the dotted line is y equals negative 2 x plus 3. The lines intersect at left parenthesis negative 1 comma 5 right parenthesis which is not plotted as a point.
Estimate the solution to the system of these linear equations based on the graph.
y=5x+10
y=−2x+3
(1 point)
Responses
(5,−1)
left parenthesis 5 comma negative 1 right parenthesis
(1, 5)
left parenthesis 1 comma 5 right parenthesis
(−1, 5)
left parenthesis negative 1 comma 5 right parenthesis
(−1,−5)
left parenthesis negative 1 comma negative 5 right parenthesis
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(-1, 5)
What is the last step to solving a system of equations?(1 point)
Responses
Check the answer.
Check the answer.
Make a table of solutions of the linear equations.
Make a table of solutions of the linear equations.
Estimate the intersection point.
Estimate the intersection point.
Graph the lines.
Graph the lines.
Check the answer.
Use the image to answer the question.
An illustration of a coordinate plane with four quadrants shows a solid line and a dotted line with arrows at both the ends intersecting with each other. The x axis extends from negative 10 to 10 in increments of 1. The y axis extends from negative 5 to 15 in increments of 1. The equation of the solid line is y equals negative 2 x plus 10. The equation of the dotted line is y equals negative 5 x plus 7. The lines intersect at left parenthesis negative 1 comma 12 right parenthesis which is not plotted as a point.
Estimate the solution to the system of these linear equations based on the graph.
y=−5x+7
y=−2x+10(1 point)
Responses
(−1, 12)
left parenthesis negative 1 comma 12 right parenthesis
(12,−1)
left parenthesis 12 comma negative 1 right parenthesis
(−1,−12)
left parenthesis negative 1 comma negative 12 right parenthesis
(1,−12)
left parenthesis 1 comma negative 12 right parenthesis
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(-1, 12)
To find the coordinates of the intersection point that solves the system of equations, you need to find the point where the two lines intersect on the coordinate plane.
1. Start by graphing the equation y = -x + 5. This is a dotted line on the coordinate plane. Since the equation is already in slope-intercept form (y = mx + b), you can easily identify the y-intercept (5) and the slope (-1). Plot the y-intercept on the y-axis, and then use the slope to find additional points on the line. Draw a dotted line through these points.
2. Next, graph the equation 2x + y = 15. To do this, rewrite the equation in slope-intercept form: y = -2x + 15. Again, identify the y-intercept (15) and the slope (-2). Plot the y-intercept on the y-axis, and then use the slope to find additional points on the line. Draw a solid line through these points.
3. The coordinate where the two lines intersect is the solution to the system of equations. In this case, the lines intersect at positive 10 on the x-axis and negative 5 on the y-axis in quadrant 4.
Therefore, the coordinates of the intersection point are (10, -5).
So, the correct answer is:
(10, -5)