Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

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Algebra 1

2.6.1 Multiplying Equations by a Number

Algebra 12.6.1 Multiplying Equations by a Number

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Warm Up

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

Consider two equations in a system:

${\begin{matrix} 4 x + y = 1 & Equation A \\ x + 2 y = 9 & Equation B \end{matrix}$

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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What is the $x$ value?

-1

What is the $y$ value?

For 4 - 6, use the following equation:

Equation A: $4 x + y = 1$

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

Compare your answer:

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

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

Compare your answer:

$- 5 (4 x + y = 1) = - 20 x - 5 y = - 5$

Multiply the equation by $\frac{1}{2},$ show the resulting equivalent equation, and label it A3.

Compare your answer:

$\frac{1}{2} (4 x + y = 1) = 2 x + \frac{1}{2} y = \frac{1}{2}$

For 7 - 9, use the following equations:

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

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.

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:

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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:

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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 $(- 1, 5)$ .

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Citation information

Use the information below to generate a citation. We recommend using a citation tool such as this one.
- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
- Publisher/website: OpenStax
- Book title: Algebra 1
- Publication date: Jun 4, 2025
- Location: Houston, Texas
- Book URL: https://openstax.org/books/algebra-1/pages/about-this-course
- Section URL: https://openstax.org/books/algebra-1/pages/2-6-1-multiplying-equations-by-a-number

© May 21, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.