standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Activity
The mathematical purpose for this activity is to allow students to practice solving for a variable in an equation with two variables. In the associated Algebra lesson, expressing one variable in terms of another is necessary before substituting an expression for a variable in another equation.
Launch
Demonstrate how to solve for a variable by showing students the equation
\(a + 2b = 5\)
. Solving for
\(a\)
results in
\(a = 5 - 2b\)
or
\(a = \text{-}2b + 5\)
. Solving for
\(b\)
results in
\(b = \frac{5-a}{2}\)
or
\(b = \frac{5}{2} - \frac{a}{2}\)
or another equivalent expression.
Student Facing
Solve for the indicated variable.
Solve for
\(k\)
.
\(2t+k=6\)
Solve for
\(n\)
.
\(10n=2p\)
Solve for
\(c\)
.
\(12-6d=3c\)
Solve for
\(g\)
.
\(h=8g+4\)
Solve for
\(x\)
.
\(4x+3y=12\)
Solve for
\(y\)
.
\(4x+3y=12\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The purpose of the discussion is to share strategies for solving for one variable in an equation including more than one variable. Select students to share their answers and strategies. As students share, consider asking another student why each step makes sense to do.
Some questions for discussion:
"How do you know when you are done solving for a variable?" (When the variable I want to solve for is the only thing on one side of the equation.)
"How is solving for
\(x\)
in an equation like
\(4x+3y=12\)
similar or different from solving for
\(x\)
in an equation like
\(4x + 6 = 12\)
?" (The steps are very similar, but when there is only one variable, we can know the value of
\(x\)
that makes the equation true.)
"If you know the value for
\(y\)
is 2 and need to know the value for
\(x\)
, would you rather begin with the equation
\(4x + 3y = 12\)
or the equivalent equation
\(x = 3 - \frac{3}{4}y\)
? Explain your reasoning." (I would rather use the second equation since I only need to substitute in the value for
\(y\)
and evaluate the expression rather than solve an equation.)


aligns: no