Linear equations in two variables

Learning target:
Students use a table to find solutions to a given linear equation and plot
the solutions on a coordinate plane.

• We discussed U3L2: Constant Rate yesterday
• Did you complete the practice problems and submit your score
to the dropbox?

In this lesson, you will find solutions to a linear equation in two variables
using a table, then plot the solutions as points on the coordinate plane. You
will need graph paper in order to complete the Exercises and the Problem Set.

Emily tells you that she scored 32 points in a basketball game with only two-
and three-point baskets (no free throws). How many of each type of basket did
she score? Use the table below to organize your work.
Number of two-pointers Number of three-pointers

Let x be the number of two-pointers and y be the number of three-pointers
that Emily scored. Write an equation to represent the situation.

An equation in the form of ax + by = c is called a linear equation in two
variables, where a, b, and c are constants, and at least one of a and b are not
zero. In this lesson, neither a nor b will be equal to zero.
In the Opening Exercise, what equation did you write to represent Emily’s score
at the basketball game?

An equation of this form, ax + by = c, is also referred to as an equation in
standard form. Is the equation you wrote for the example with Emily an
equation in standard form?

In the equation ax + by = c, the symbols a, b, and c are constants. What, then,
are x and y?

For example, −50x + y = 15 is a linear equation in x and y. As you can easily see,
not just any pair of numbers x and y will make the equation true. Consider x = 1
and y = 2. Does it make the equation true?

What pairs of numbers did you find that worked for Emily’s basketball score?
Did just any pair of numbers work? Explain.

A solution to the linear equation in two variables is an ordered pair of numbers
(x, y) so that x and y makes the equation a true statement. The pairs of numbers
that you wrote in the table for Emily are solutions to the equation 2x + 3y = 32
because they are pairs of numbers that make the equation true. The question
becomes, how do we find an unlimited number of solutions to a given linear
equation?

A strategy that will help us find solutions to a linear equation in two variables is
as follows: We fix a number for x. That means we pick any number we want and
call it x. Since we know how to solve a linear equation in one variable, then we
solve for y. The number we picked for x and the number we get when we solve
for y is the ordered pair (x, y), which is a solution to the two variable linear
equation.
For example, let x = 5. Then, in the equation −50x + y = 15

Similarly, we can fix a number for y and solve for x. Let y = 10, then
-50x + y = 15

Now you pick a number for x or y and find the solution to the equation
-50x + y = 15

Find five solutions for the linear equation x + y = 3, and plot the solutions as
points on a coordinate plane.
x Linear equation:
x + y = 3
y

Linear equations in two variables

Find five solutions for the linear equation 2x − y = 10, and plot the solutions as
points on a coordinate plane.
x Linear equation:
2x – y = 10
y

Find five solutions for the linear equation x + 5y = 21, and plot the solutions
as points on a coordinate plane.
x Linear equation:
x + 5y = 21
y

Consider the linear equation
2
5
x + y = 11.
Will you choose to fix values for x or y? Explain
Are there specific numbers that would make your computational work easier?
Explain

Find five solutions to the linear equation
2
5
x + y = 11, and plot the solutions as
points on the coordinate plane.
x Linear equation:
𝟐
𝟓
x + y = 11
y

At the store, you see that you can buy a bag of candy for $2 and a drink for $1.
Assume you have a total of $35 to spend. You are feeling generous and want to
buy some snacks for you and your friends. a. Write an equation in standard
form to represent the number of bags of candy, x, and the number of drinks, y,
that you can buy with $35.

Find five solutions to the linear equation from the first part, and plot the
solutions as points on a coordinate plane.
x Linear equation: y

• A two-variable equation in the form of ax + by = c is known as a linear
equation in standard form.
• A solution to a linear equation in two variables is an ordered pair (x, y) that
makes the given equation true.
• We can find solutions by fixing a number for x or y, then solving for the
other variable. Our work can be made easier by thinking about the
computations we will need to make before fixing a number for x or y. For
example, if x has a coefficient of
1
3
, we should select values for x that are
multiples of 3.

• Complete the U3L3 practice problems, check your answers by watching
the recording, and then submit your score to the dropbox.

Linear equations in two variables

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