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Multiplying Equations by a Number

2.6.1 Multiplying Equations by a Number

2.6.1 • Multiplying Equations by a Number

Warm Up

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Consider two equations in a system:

{ 4x+y=1Equation A x+2y=9Equation B { 4x+y=1Equation A x+2y=9Equation B

1.

Use the graphing tool or technology outside the course. Graph the system of equations using the Desmos tool below.

Compare your answer:

Graph showing two intersecting lines: a blue line with a gentle, negative slope passing through (9, 0) and a black line with a steep, negative slope passing through (0, 1), on an x-y coordinate plane.
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2.

What is the xx value?

-1

3.

What is the yy value?

5

For questions 4 – 6, use the following equation:

Equation A: 4x+y=14x+y=1
4.

Multiply the equation by 2, show the resulting equivalent equation, and label it A1.

Compare your answer:

2(4x+y=1)=8x+2y=22(4x+y=1)=8x+2y=2

5.

Multiply the equation by -5, show the resulting equivalent equation, and label it A2.

Compare your answer:

−5(4x+y=1)=−20x−5y=−5−5(4x+y=1)=−20x−5y=−5

6.

Multiply the equation by 12,12, show the resulting equivalent equation, and label it A3.

Compare your answer:

12(4x+y=1)=2x+12y=1212(4x+y=1)=2x+12y=12

For questions 7 – 9, use the following equations:

The new equations A1, A2, and A3 and Equation B: x+2y=9x+2y=9
7.

Use the graphing tool or technology outside the course. Graph the system of equations A1 and B using the Desmos tool below.

Compare your answer:

Graph of a system of equations. The orange line represents an equation with a steeply negative slope. The green line also has a negative slope but is less steep. The lines intersect in the third quadrant.
8.

Use the graphing tool or technology outside the course. Graph the system of equations A2 and B using the Desmos tool below.

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Compare your answer:

Graph of a system of equations. The orange line represents an equation with a steeply negative slope. The green line also has a negative slope but is less steep. The lines intersect in the third quadrant.
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9.

Use the graphing tool or technology outside the course. Graph the system of equations A3 and B using the Desmos tool below.

Compare your answer:

Graph of a system of equations. The orange line represents an equation with a steeply negative slope. The green line also has a negative slope but is less steep. The lines intersect in the third quadrant.
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10.

Make a couple of observations about the graphs of Equations A1, A2, and A3.

Compare your answer:

Your answer may vary, but here is a sample. No new graphs appeared. The graphs of the equivalent equations are identical to that of Equation A. The graphs all intersect the graph of Equation B at (−1,5)(−1,5).