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Algebra 1

1.11.3 Reasoning Symbolically and Abstractly about Linear Equations

Algebra 11.11.3 Reasoning Symbolically and Abstractly about Linear Equations

1.11.3 Reasoning Symbolically and Abstractly about Linear Equations

1.11.3 • Reasoning Symbolically and Abstractly about Linear Equations

Activity

For questions 1 – 5, select the slope and intercept that corresponds to each equation:

1.

− 4 x + 3 y = 3 − 4 x + 3 y = 3

  1. m = 3 m = 3 , y -int = ( 0 , 1 ) y -int = ( 0 , 1 )

  2. m = 4 3 m = 4 3 , y -int = ( 0 , 1 ) y -int = ( 0 , 1 )

  3. m = 4 3 m = 4 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

  4. m = − 4 m = − 4 , y -int = ( 0 , 8 ) y -int = ( 0 , 8 )

  5. m = 3 m = 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

2.

12 x − 4 y = 8 12 x − 4 y = 8

  1. m = 3 m = 3 , y -int = ( 0 , 1 ) y -int = ( 0 , 1 )

  2. m = 4 3 m = 4 3 , y -int = ( 0 , 1 ) y -int = ( 0 , 1 )

  3. m = 4 3 m = 4 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

  4. m = − 4 m = − 4 , y -int = ( 0 , 8 ) y -int = ( 0 , 8 )

  5. m = 3 m = 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

3.

8 x + 2 y = 16 8 x + 2 y = 16

  1. m = 3 m = 3 , y -int = ( 0 , 1 ) y -int = ( 0 , 1 )

  2. m = 4 3 m = 4 3 , y -int = ( 0 , 1 ) y -int = ( 0 , 1 )

  3. m = 4 3 m = 4 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

  4. m = − 4 m = − 4 , y -int = ( 0 , 8 ) y -int = ( 0 , 8 )

  5. m = 3 m = 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

4.

− x + 1 3 y = 1 3 − x + 1 3 y = 1 3

  1. m = 3 , y -int = ( 0 , 1 ) m = 3 , y -int = ( 0 , 1 )

  2. m = 4 3 m = 4 3 , y -int = ( 0 , 1 ) y -int = ( 0 , 1 )

  3. m = 4 3 m = 4 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

  4. m = − 4 m = − 4 , y -int = ( 0 , 8 ) y -int = ( 0 , 8 )

  5. m = 3 m = 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

5.

− 4 x + 3 y = − 6 − 4 x + 3 y = − 6

  1. m = 3 m = 3 , y -int = ( 0 , 1 ) y -int = ( 0 , 1 )

  2. m = 4 3 m = 4 3 , y -int = ( 0 , 1 ) y -int = ( 0 , 1 )

  3. m = 4 3 m = 4 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

  4. m = − 4 m = − 4 , y -int = ( 0 , 8 ) y -int = ( 0 , 8 )

  5. m = 3 m = 3 , y -int = ( 0 , − 2 ) y -int = ( 0 , − 2 )

Using Standard Form and Intercepts to Graph

Standard form, A x + B y = C A x + B y = C , is helpful for finding the x x -intercepts and y y -intercepts of an equation.

For questions 6 – 11, use the equation 6 x − 4 y = 24 6 x − 4 y = 24 .

6.

What is the value of y y at the x x -intercept?

7.

What is the x x -intercept of the equation?

8.

What is the value of x x at the y y -intercept?

9.

What is the y y -intercept of the equation?

10.

Use the graphing tool or technology outside the course. Plot the two intercepts and the equation of the line, using the Desmos tool below.

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11.

Using the graph from question 10, what is the slope of the line in simplest form?

Are you ready for more?

Extending Your Thinking

For questions 1-2, use the equations below:

These are the equations from the activity. Each equation is in the form A x + B y = C A x + B y = C .
 4 x − 3 y = − 3 4 x − 3 y = − 3
12 x − 4 y = 8 12 x − 4 y = 8
8 x + 2 y = 16 8 x + 2 y = 16
3 x − y = − 1 3 x − y = − 1
4 x − 3 y = 6 4 x − 3 y = 6
1.

Observe the graph of the equation − 4 x + 3 y = 3 − 4 x + 3 y = 3 where it is graphed, on the same coordinate plane, with another line passing through ( 0 , 0 ) ( 0 , 0 ) and ( A , B ) ( A , B ) . In this example A = − 4 A = − 4 and B = 3 B = 3 , so the other line would pass through ( 0 , 0 ) ( 0 , 0 ) and ( − 4 , 3 ) ( − 4 , 3 ) . What do you notice about the lines?

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2.

Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool above. Then make another observation about the lines.

Self Check

What is the slope of the line 4 x + 3 y = 12 ?
  1. 4 3
  2. − 3 4
  3. 3 4
  4. − 4 3

Additional Resources

Video: Using Standard Form

Watch the following video to learn more about using standard form to find slope and y y -intercepts.

Using Standard Form

Try it

Using Standard Form

1.

Find the slope for 5 x − y = 15 5 x − y = 15 .

2.

Find the y y -intercept for 5 x − y = 15 5 x − y = 15 .

3.

Explain your reasoning.

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