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 (10,−5) left parenthesis 10 comma negative 5 right parenthesis (5,−10) left parenthesis 5 comma negative 10 right parenthesis (−10, 5) left parenthesis negative 10 comma 5 right parenthesis (−5, 10)
The coordinates of the intersection point that solves the system of the two linear equations are (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 . 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 intersect at (−2, 6). Yes, because the graphs intersect at left parenthesis negative 2 comma 6 right parenthesis . Yes, because the graphs don’t intersect at (−2, 6)
To check if (-2, 6) is a solution to the system of linear equations, we substitute the x and y values into both equations:
For the equation x + 2y = 10:
(-2) + 2(6) = 10
-2 + 12 = 10
10 = 10 (True)
For the equation 3x + y = 0:
3(-2) + 6 = 0
-6 + 6 = 0
0 = 0 (True)
Since both equations are true when we substitute (-2, 6) into them, (-2, 6) is indeed a solution to the system of linear equations. Therefore, the correct response is: Yes, because the graphs intersect at (-2, 6).
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 negative 1 comma negative 5 right parenthesis (−1, 5) left parenthesis negative 1 comma 5 right parenthesis (1, 5)
Based on the graph, the estimate for the solution to the system of linear equations is (-1, 5).
What is the last step to solving a system of equations?(1 point) Responses Graph the lines. Graph the lines. 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.
The last step to solving a system of equations is to check the answer.
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 (−1,−12) left parenthesis negative 1 comma negative 12 right parenthesis (12,−1) left parenthesis 12 comma negative 1 right parenthesis (1,−12)
Based on the graph, the estimate for the solution to the system of linear equations is (-1, 12).
Multiple Choice Question
A student is writing a report and wants to edit it to make better transitions among the paragraphs. Read the students report and answer the question that follows.
President Abraham Lincoln delivered the Gettysburg Address in 1863 at the dedication of the Soldiers' National Cemetery in Pennsylvania. The first speaker at the dedication, Edward Everett, orated for more than two hours. President Lincoln spoke after him, and his speech lasted little more than two minutes. Afterward, Everett told Lincoln that he wished he could express in two hours what Lincoln had said in two minutes.
Lincoln had delivered one of the most memorable speeches in history. This was fitting, as over 50,000 soldiers had died at the Battle of Gettysburg, a turning point in the Civil War. Lincoln's speech focused on a "new birth of freedom" in the United States, as he invoked the Declaration of Independence and its principles of liberty and equality. He stated clearly the need for the nation to be both united and free.
The oldest copies of the speech are different from one another. The Library of Congress has two copies, and the others were recorded by Lincoln well after the speech. One copy was made for Everett, but "Bancroft" and "Bliss" versions also exist. The "Bliss" version is carved into a wall of the Lincoln Memorial in our nation's capital.
Choose the sentence that provides the BEST transition between paragraphs 2 and 3.
A.
It is not exactly certain which words Lincoln spoke that day.
B.
The spirit of Lincoln's speech is remembered well today.
C.
Lincoln was certain about his vision for the course of the nation.
D.
Lincoln's speech had surprised the crowd.