How to Solve Linear Systems of Equations by Substitution

Concept

In mathematics, a system of equations, also known as a set of simultaneous or an equation system, is a finite set of equations for which we sought the common solutions. A system of equations can be classified in a similar manner as single equations.

Any system of equations can be solved in different methods. To solve a system of equations in 2 variables, we need at least 2 equations. Similarly, for solving a system of equations in 3 variables, we will require at least 3 equations. Let us understand 3 ways to solve a system of equations given the equations are linear equations in two variables.

Substitution Method
Elimination Method
Graphical Method

For solving the system of equations using the substitution method given two linear equations in x and y, express y in terms x in one of the equations and then substitute it in 2nd equation.

Rules

Step 1: Solve one of the equations for one of the variables.

Step 2: Substitute the expression of the isolated variable in the other linear equation.

Step 3: Solve the equation, and you have one of the coordinates of the intersection.

Step 4: Then plug in the value of the coordinate found in step 3 to either equation to find the other coordinate.

Example

Solve this system of equations using the substitution method.
$\fn_phv -7x-2y=-13$
$\fn_phv x-2y=11$

Solution

First, make one equation equal to a variable. It is easiest to make the second equation equal to x. Then, rewrite the first equation replacing the variable.
$\fn_phv x - 2y = 11$
$\fn_phv x = 11 + 2y$

Substitute the value of x in the other linear equation and solve it.
$\fn_phv -7x - 2y = -13$
$\fn_phv -7(11 + 2y) - 2y = -13$
$\fn_phv -77 - 14y - 2y = -13$
$\fn_phv -77 - 16y = -13$
$\fn_phv -16y = 64$
$\fn_phv y = -4$

Then, plug that value into one of the original equations.
$\fn_phv x = 11 + 2y$
$\fn_phv x = 11 + 2(-4)$
$\fn_phv x = 11 - 8$
$\fn_phv x = 3$

The solution is (3, -4)

Practice How to Solve Linear Systems of Equations by Substitution

Practice Problem 1

Solve this system of equations using the substitution method.
$\fn_phv x - y = 3$
$\fn_phv y = 3x - 1$

Practice Problem 2

Solve this system of equations using the substitution method.
$\fn_phv 4x - 5y = -23$
$\fn_phv 2x + y = -1$

Practice Problem 3

A movie theater charges $5 for adults and $3 for children. Last Friday, 700 tickets were sold to the public totaling $2,900. Given the two equations:
$\fn_phv a + c = 700$
$\fn_phv 5a + 3c = 2,900$
Find how many adults and how many children attended the movie.

A system of linear equations – is a set of equations which are satisfied by the same set of variables.

Variable – A symbol, usually a letter, used to represent a number in mathematical expressions or sentences.

Equation – A mathematical sentence stating that two quantities are equal.

Slope intercept form – a linear equation written in the form y = mx + b, where m stands for the slope, and b stands for the y-intercept.

Standard form – a linear equation written in the form Ax + By = C

Solution – A solution to a system of equations is the value of the variables that makes ALL the equations True.

Substitution – Putting numbers in place of letters to calculate the value of an expression.

Pre-requisite Skills
Solve Equations with Rational Coefficients
Solve Two-Step Equations
Write Two-Step Equations
Solve Two-Sided Equations with Rational Coefficients
Solve Multi-Step Equations
Slope-intercept Form of an Equation
Graphing Lines Using Intercepts
Write Linear Equations
Direct Variation Equations (y = mx)
Solutions of Systems of Equations

Geometry and Spatial sense

Graphs

How to Solve Linear Systems of Equations by Substitution

How to Solve Linear Systems of Equations by Substitution

Practice Problem 1

Practice Problem 2

Practice Problem 3

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