Lesson 14

Making More New, True Equations

These materials, when encountered before Algebra 1, Unit 2, Lesson 14 support success in that lesson.

14.1: Criss Cross'll Make You Jump (5 minutes)

Warm-up

In this activity students work to understand the form of equations that will result in vertical or horizontal lines. In later activities in the lesson, students will be asked to write equations of horizontal or vertical lines that have certain properties.

Student Facing

Match each equation with its graph.

  • \(x=7\)
  • \(y=7\)
  • \(x+y=7\)

A

Y=7 graphed on Coordinate plane 

B

Graph of x=7 on coordinate plane 

C

Line graphed on coordinate plane. X intercept = 7. Y intercept =7. 

D

Graph of line. Passes through -2 comma-10 and origin 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of the discussion is to ensure students understand that, for a constant \(a\), the equation of the form \(x = a\) is represented by a vertical line and the equation of the form \(y=a\) is represented by a horizontal line.

Some questions for discussion:

  • "Why is an equation like \(y = 7\) represented by a horizontal line?" (Because all of the coordinate pairs with a \(y\)-value of 7 line up horizontally.)
  • "How can you know from the equation alone whether it will be represented by a horizontal, vertical, or tilted line?" (Horizontal lines are all of the form \(y=a\), vertical lines are all of the form \(x=a\) , and tilted lines will have both \(x\) and \(y\) in the equation (with nonzero coefficients).)

14.2: They're Like Terms, Man (20 minutes)

Activity

The purpose of this activity is for students to practice combining like terms for expressions similar to what students will see when solving systems of equations by elimination. Students will need to add or subtract expressions to eliminate a variable. This activity gives students the chance to practice that skill.

Launch

Remind students what the phrase "combining like terms" means by showing that the example \(3x + 5x\) results in \(8x\)

Student Facing

Rewrite each expression by combining like terms.

  1. \(11s-2s\)

  2. \(5t+3z-2t\)

  3. \(23s-(13t+7t)\)

  4. \(7t + 18r + (2r - 5t)\)
  5. \(\text{-}4x + 6r - (7x + 2r)\)
  6. \(3(c-5) + 2c\)
  7. \(8x - 3y + (3y - 5x)\)
  8. \(5x + 4y - (5x + 7y)\)
  9. \(9x - 2y - 3(3x+y)\)
  10. \(6x+12y + 2(3x-6y)\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Discuss how combining like terms in expressions can be useful in solving equations. Select students to share their solutions including any intermediate steps they used to arrive at their answer. Some questions for discussion:

  • "Would you rather use the expression \(3x + 4y - (5x - 2y)\) or the equivalent expression \(\text{-}2x + 6y\) to find the value when you know \(x = 1\) and \(y = 3\)?" (The shorter expression that has combined like terms would be simpler to use.)
  • "Use the values \(c = 1, r = 2, t = 3, x = 4, y = 5\) to find the value of the original expressions and the value of your answers. What do you notice? Why do you think that is?" (The values of both expressions are the same since they expressions are equivalent.)

14.3: Finding More Lines (15 minutes)

Activity

In this activity, students solve a system of equations by graphing then include a third line through the solution point that is either vertical or horizontal. In the associated Algebra lesson, students will graph equations in a system and additional equations that arise from the elimination method to show that they all intersect at the solution.

To focus on the work of the activity, it is recommended that students use the digital version of the activity or have access to Desmos or another graphing tool to more quickly and accurately find the solutions to the system of equations through graphing. If students need practice graphing lines representing equations by hand or if digital access is not available, the print version provides a graph given one of the lines and asks students to graph the second line by hand.

Launch

Display the instructions for all to see. Ask students,

  • "What do you know about equations that are represented by vertical lines?" (They are of the form \(x = a\) for a number \(a\).)
  • "What do you know about equations that are represented by horizontal lines?" (They are of the form \(y = a\) for a number \(a\).)

Student Facing

For each system of equations:

  • Solve the system of equations by graphing. Write the solution as an ordered pair.
  • Write an equation that would represented by a vertical or horizontal line that also passes through the solution of the system of equations.
  • Graph your new equation along with the system.
  1. \(\begin {cases} \begin {align}y = 3x+5\\ y=\text{-}x+1\end{align} \end {cases}\)

    The line representing $y = 3x+5$ is shown

    Graph of y=3x +5
  2. \(\begin {cases} \begin {align}y = \frac{1}{3}x-2\\ y=x-6\end{align} \end {cases}\)

    The line representing $y = \frac{1}{3}x-2$ is shown

    Graph of y = one third x - 2
  3. \(\begin {cases} \begin {align} 2x+3y=10\\ x+y=3\end{align} \end {cases}\)

    The line representing $2x+3y=10$ is shown

    Graph of 2x -3y =10

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of the discussion is for students to recognize that, for a system of equations represented by graphs that intersect at \((a,b)\), the lines \(x = a\) and \(y = b\) also go through the solution. Select students to share their solutions including students who found vertical as well as horizontal lines through the intersection point.

Some questions for discussion:

  • "How could you use your answers to these questions to write another system of equations that have the same solution as the original system of equations?" (The system \(\begin {cases} \begin {align} x = \text{-}1\\ y=2\end{align} \end {cases}\) has the same solution as the original equation \(\begin {cases} \begin {align}y = 3x+5\\ y=\text{-}x+1\end{align} \end {cases}\) since all 4 lines go through the point \((\text{-}1,2)\).)
  • "Two lines intersect at the point \((3,5)\). How could you know whether the equation \(3x - 2y = \text{-}1\) is represented by a line that goes through that same point?" (Substitute the value 3 for \(x\) and 5 for \(y\). If the equation is still true, then the associated line will go through the same point. If the equation is false, the associated line will not go through the same point.)