Lesson 11More Solutions to Linear Equations

Learning Goal

Let’s find solutions to more linear equations.

Learning Targets

  • I can find solutions to linear equations given either the - or the -value to start from.

Lesson Terms

  • solution to an equation with two variables

Warm Up: Coordinate Pairs

Problem 1

For each equation choose a value for and then solve to find the corresponding value that makes that equation true.

Activity 1: True or False: Solutions in the Coordinate Plane

Problem 1

Here are graphs representing three linear relationships. These relationships could also be represented with equations.

A graph with lines l, m, and n. Line l has point H and A. Line m has points D, G, K. Line n has points G, E, H.

For each statement below, decide if it is true or false. Explain your reasoning.

After you finish discussing the eight statements, find another group and check your answers against theirs. Discuss any disagreements.

  1. is a solution of the equation for line .

  2. The coordinates of the point make both the equation for line and the equation for line true.

  3. is a solution of the equation for line .

  4. makes both the equation for line and the equation for line true.

  5. There is no solution for the equation for line that has .

  6. The coordinates of point are solutions to the equation for line .

  7. There are exactly two solutions of the equation for line .

  8. There is a point whose coordinates make the equations of all three lines true.

Activity 2: I’ll Take an X, Please

Problem 1

One partner has 6 cards labeled A through F and one partner has 6 cards labeled a through f. In each pair of cards (for example, Cards A and a), there is an equation on one card and a coordinate pair, , that makes the equation true on the other card.

  1. The partner with the equation asks the partner with a solution for either the -value or the -value and explains why they chose the one they did.

  2. The partner with the equation uses this value to find the other value, explaining each step as they go.

  3. The partner with the coordinate pair then tells the partner with the equation if they are right or wrong. If they are wrong, both partners should look through the steps to find and correct any errors. If they are right, both partners move onto the next set of cards.

  4. Keep playing until you have finished Cards A through F.

Are you ready for more?

Problem 1

Consider the equation , where and are positive numbers.

  1. Find the coordinates of the - and -intercepts of the graph of the equation.

  2. Find the slope of the graph.

Activity 3: Making Signs

Problem 1

Clare and Andre are making signs for all the lockers as part of the decorations for the upcoming spirit week. Yesterday, Andre made 15 signs and Clare made 5 signs. Today, they need to make more signs. Each person’s progress today is shown in the coordinate plane.

A graph of time in minutes vs number of completed signs. Blue line starts at (0,5) with points A and B. Black line C starts at (0,15) and intersects the blue line at A.

Based on the lines, mark the statements as true or false for each person.

point

what it says

Clare

Andre

At 40 minutes, I have 25 signs completed.

At 75 minutes, I have 42 and a half signs completed.

At 0 minutes, I have 15 signs completed.

At 100 minutes, I have 60 signs completed.

Are you ready for more?

Problem 1

  • 4 toothpicks make 1 square

  • 7 toothpicks make 2 squares

  • 10 toothpicks make 3 squares

Three drawings - a square, two connected squares, and three connected squares.

Do you see a pattern? If so, how many toothpicks would you need to make 10 squares according to your pattern? Can you represent your pattern with an expression?

Lesson Summary

Let’s think about the linear equation . If we know  is a solution to the equation, then we also know  is a point on the graph of the equation. Since this point is on the -axis, we also know that it is the vertical intercept of the graph. But what about the coordinate of the horizontal intercept, when ? Well, we can use the equation to figure it out.

Since when, we know the point is on the graph of the line. No matter the form a linear equation comes in, we can always find solutions to the equation by starting with one value and then solving for the other value.