## Two-step Linear Equations with Rational Numbers

Materials: Large grid paper or transparent grid for overhead projector (this grid should go at least from 10 to 10 on both axes), grease pencil or marker, ruler or yardstick

• Say When we generated points for lines yesterday, our equations were always in the same form. Today, we'll look at other linear equations.

• Say Can someone describe how to find some coordinate pairs for the linear equation, 2x + y = 15?
This is a two-step equation. Solutions involve assigning a value to x, then multiplying this value by 2 before trying to figure out what value of y would satisfy the equation. Students can use trial and error, or they can transform the equation using the equality properties:

 Write the equation. 2x + y = 15 Assign a value to x. 2(3) + y = 15 Multiply. 6 + y = 15 Subtract 6 from each side. 6 – 6 + y = 15 – 6 y = 9
This solution gives us the point (3, 9)

Continue finding solutions (coordinate pairs) for this equation until you are satisfied that students are comfortable with the process. Then, plot the points on your grid and draw the line.

• Say Can someone describe how to find some points on the line described by the equation, y + x/3 = 5?

 Write the equation. y + x/3 = 5 Assign a value to x. y + 3/3 = 5 Divide. y + 1 = 5 Subtract 1 from each side. y + 1 – 1 = 5 – 1 y = 4

This solution gives us the point (3, 4)

Continue finding solutions (coordinate pairs) for this equation until you are satisfied that students are comfortable with the process. Then, plot the points on your grid and draw the line.

• Say Can someone describe how to find some points on the line described by the equation, y – 6 = 2x?

 Write the equation y – 6 = 2x Assign a value to x. y – 6 = 2(3) Multiply. y – 6 = 6 Add 6 to each side. y – 6 + 6 = 6 + 6 y = 12

This solution gives us the point (3, 12)

Continue finding solutions (coordinate pairs) for this equation until you are satisfied that students are comfortable with the process. Then, plot the points on your grid and draw the line.

• By now, students should have noticed that the easiest substitution for x is 0. This substitution will give you the point where the line crosses the y axis. If this realization has not come up independently, prompt for it.

Wrap-Up and Assessment Hints
When students are solving multi-step equations, pay particular attention to whether they follow the order of operations. This is an important algebraic concept. Also, watch for whether students really understand that the equality properties say that if you do something to one side of an equation, you MUST do the same thing to the other side of the equation. What you do is determined by the action indicated by the equation. If a number is subtracted from y and you want y to be by itself, add that number to each side of the equation and its opposite appears to 'move' to the other side of the equation. Similarly, if y is multiplied by a number, division will help you get y by itself.

Slope        Two-step Linear Equations
with Rational Numbers.