Solving Systems of Equations by Substitution

Systems of equations are simultaneous equations that share the same solutions. We can solve systems of two equations with two unknowns using the substitution method. This method consists of solving for one of the variables in one of the equations. Then we need to substitute that expression into the second equation to form a single equation with one unknown.

Here, we will learn how to solve systems of equations using the substitution method. We will solve several practice problems to understand the process used.

ALGEBRA
linear systems of equations

Relevant for

Learning to solve systems of equations with the substitution method.

See method

ALGEBRA
linear systems of equations

Relevant for

Learning to solve systems of equations with the substitution method.

See method

Steps to solve systems of equations by substitution

The substitution method for solving systems of equations consists of solving one of the equations for one of the variables. Then, we substitute the obtained expression in the second equation, and we will obtain an equation with a single variable, which can be solved easily.

In detail, we can follow the following steps to solve systems of equations by substitution:

Step 1: Simplify both equations if possible. This includes removing parentheses, combining like terms, and removing fractions.

Step 2: Solve an equation for one variable. We can choose any of the equations and solve for any of the variables.

Step 3: Substitute the expression obtained in step 2 in the other equation. By doing this, we will obtain an equation with only one unknown.

Step 4: Solve the equation from step 3. By doing this, we will get the value of one of the variables.

If you need to review, you can take a look at our article on how to solve linear equations with one variable.

Step 5: Find the value of the other variable. We substitute the value of the variable obtained in step 4 in any of the two equations and solve.

Step 6: Check the solution in both equations. A solution to a system of equations must satisfy all the equations in the system.


Systems of equations by substitution – Examples with answers

In the following examples, we have to find the solutions to the systems of equations using the substitution method. Try to solve the problems yourself before looking at the solution.

EXAMPLE 1

Find the solutions to the system of equations using the substitution method: $latex \begin{cases}x+2y=10 \\ 2x-y=5 \end{cases}$

Step 1: The equations are already simplified.

Step 2: Solving the first equation for x, we have:

$latex x+2y=10$

$latex x=10-2y$

Step 3: Substituting $latex for x=10-2y$ in the second equation, we have:

$latex 2x-y=5$

$latex 2(10-2y)-y=5$

$latex 20-4y-y=5$

Step 4: Solving for y, we have:

$latex 20-4y-y=5$

$latex -5y=-15$

$latex y=3$

Step 5: Substituting the value $latex y=3$ in the first equation, we have:

$latex x+2y=10$

$latex x+2(3)=10$

$latex x=4$

The solution to the system is $latex x=4,~~y=3$.

EXAMPLE 2

Solve the following system of equations using the substitution method: $latex \begin{cases}-2x-y=1 \\ 3x+4y=6 \end{cases}$

Step 1: The equations are already simplified.

Step 2: Solving the first equation for y, we have:

$latex -2x-y=1$

$latex -y=1+2x$

$latex y=-1-2x$

Step 3: Substituting $latex y=-1-2x$ into the second equation, we have:

$latex 3x+4y=6$

$latex 3x+4(-1-2x)=6$

$latex 3x-4-8x=6$

Step 4: Solving for x, we have:

$latex 3x-4-8x=6$

$latex -5x=10$

$latex x=-2$

Step 5: Substituting $latex x=-2$ into the first equation, we have:

$latex -2x-y=1$

$latex -2(-2)-y=1$

$latex 4-y=1$

$latex -y=-3$

$latex y=3$

The solution to the system is $latex x=-2, ~~y=3$.

EXAMPLE 3

Find the solution to the system of equations using the substitution method: $latex \begin{cases}2(2x-4)+y=3 \\ -x+2y=4 \end{cases}$

Step 1: We can simplify the first equation:

$latex \begin{cases}4x-8+y=3 \\ -x+2y=4 \end{cases}$

Step 2: Solving the first equation for y, we have:

$latex 4x-8+y=3$

$latex y=-4x+11$

Step 3: Substituting $latex y=-4x+11$ into the second equation, we have:

$latex -x+2y=4$

$latex -x+2(-4x+11)=4$

$latex -x-8x+22=4$

Step 4: Solving for x, we have:

$latex -x-8x+22=4$

$latex -9x=-18$

$latex x=2$

Step 5: Substituting $latex x=2$ into the second equation, we have:

$latex -x+2y=4$

$latex -2+2y=4$

$latex 2y=6$

$latex y=3$

The solution to the system is $latex x=2,~~y=3$.

EXAMPLE 4

Solve the system of equations: $latex \begin{cases}3x+4y-27=0 \\ 5x+y-11=0 \end{cases}$

Step 1: The equations are already simplified.

Step 2: We can solve the second equation for y:

$latex 5x+y-11=0$

$latex y=-5x+11$

Step 3: Substituting $latex y=-5x+11$ into the first equation, we have:

$latex 3x+4y-27=0$

$latex 3x+4(-5x+11)-27=0$

$latex 3x-20x+44-27=0$

$latex -17x+17=0$

Step 4: Solving for x, we have:

$latex -17x+17=0$

$latex -17x=-17$

$latex x=1$

Step 5: Substituting $latex x=1$ into the second equation, we have:

$latex 5x+y-11=0$

$latex 5(1)+y-11=0$

$latex y-6=0$

$latex y=6$

The solution to the system is $latex x=1,~~y=6$.

EXAMPLE 5

Find the solution to the system of equations: $latex \begin{cases}2(-x+y)=-3x+y+9 \\ 2x+y=13 \end{cases}$

Step 1: Simplifying the first equation, we have:

$latex \begin{cases}x+y=9 \\ 2x+y=13 \end{cases}$

Step 2: Solving the first equation for y, we have:

$latex x+y=9$

$latex y=9-x$

Step 3: Substituting $latex y=9-x$ into the second equation, we have:

$latex 2x+y=13$

$latex 2x+9-x=13$

Step 4: Solving for x, we have:

$latex 2x+9-x=13$

$latex x=4$

Step 5: Substituting $latex x=4$ into the first equation, we have:

$latex x+y=9$

$latex 4+y=9$

$latex y=5$

The system solution is $latex x=4,~~y=5$.

EXAMPLE 6

Find the solution to the system of equations: $latex \begin{cases}2x-3y=7 \\ 2x+3y=1 \end{cases}$

Step 1: The equations are already simplified.

Step 2: Solving the first equation for x, we have:

$latex 2x-3y=7$

$latex 2x=3y+7$

$latex x=\frac{3y+7}{2}$

Step 3: Substituting $latex x=\frac{3y+7}{2}$ into the second equation, we have:

$latex 2x+3y=1$

$latex 2\left(\frac{3y+7}{2}\right)+3y=1$

$latex 3y+7+3y=1$

Step 4: Solving for y, we have:

$latex 3y+7+3y=1$

$latex 6y=-6$

$latex y=-1$

Step 5: Substituting $latex y=-1$ into the second equation, we have:

$latex 2x+3y=1$

$latex 2x+3(-1)=1$

$latex 2x-3=1$

$latex 2x=4$

$latex x=2$

The solution to the system is $latex x=2,~~y=-1$.

EXAMPLE 7

Solve the system of equations using the substitution method: $latex \begin{cases}3x-4y=5 \\ 6x-4y=2 \end{cases}$

Step 1: We can simplify the second equation by dividing it by 2:

$latex \begin{cases}3x-4y=5 \\ 3x-2y=1 \end{cases}$

Step 2: Solving the second equation for x, we have:

$latex 3x-2y=1$

$latex 3x=2y+1$

$latex x=\frac{2y+1}{3}$

Step 3: Substituting $latex x=\frac{2y+1}{3}$ into the first equation, we have:

$latex 3x-4y=5$

$latex 3\left(\frac{2y+1}{3}\right)-4y=5$

$latex 2y+1-4y=5$

Step 4: Solving for y, we have:

$latex 2y+1-4y=5$

$latex -2y=4$

$latex y=-2$

Step 5: Substituting $latex y=-2$ into the second equation, we have:

$latex 3x-2y=1$

$latex 3x-2(-2)=1$

$latex 3x+4=1$

$latex 3x=-3$

$latex x=-1$

The solution to the system is $latex x=-1,~~y=-2$.


Systems of equations by substitution – Practice problems

Find the solutions to the following systems of equations using the substitution method.

Solve the system of equations: $latex \begin{cases}3x-y=1 \\ 5x+y=7 \end{cases}$

Choose an answer






Solve the system of equations: $latex \begin{cases}5x+2y=7 \\ 2x+y=2 \end{cases}$

Choose an answer






Find the solution to the system of equations: $latex \begin{cases}2x-7y=1 \\ 2x+3y=11 \end{cases}$

Choose an answer






Solve the system of equations: $latex \begin{cases}2x+3y=1 \\ 3x+y=5 \end{cases}$

Choose an answer






Solve the system of equations: $latex \begin{cases}x-2y=5 \\ 3x+y=8 \end{cases}$

Choose an answer







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