Linear systems of equations are systems that contain equations with variables of the first degree. All the equations of these systems share the same solution. Depending on their characteristics, these systems can have different types of solutions.

There are three main methods for solving linear systems of equations: by the graphical method, by the substitution method, and by the elimination method.

## Solving systems of equations graphically

We can follow the following steps to solve linear systems of equations graphically:

#### 1. Graph the first equation.

You can use any method to graph a line. If you need to, you can take a look at our guide on how to graph linear functions.

#### 2. Graph the second equation in the same coordinate system as the first.

#### 3. Find the point of intersection.

- If the lines intersect in
**only one place**, then the point of intersection is the solution to the system of equations. - If the lines are
**parallel**, then they never intersect and therefore there is no solution. - If the lines are on top of each other, then there are an infinite number of solutions.

#### 4. Check the solution to the system in both equations.

Substitute the solution into both equations. If both equations are true, then the solution is true. If either equation turns out to be false, then that ordered pair is not the correct solution.

### EXAMPLE 1

Solve the system of equations graphically: $latex \begin{cases}x+y=4 \\ 2x-y=-1 \end{cases}$

##### Solution

**Step 1:** Graph the first equation. We can rewrite the equation in the form $latex y=mx+b$, where

*m*is the slope and

*b*is the

*y*-intercept.

$latex x+y=4$

$latex y=4-x$

Therefore, the *y*-intercept is 4 and the slope is -1:

**Step 2:** Graph the second equation. We use the same method as the previous equation:

$latex 2x-y=-1$

$latex y=2x+1$

Therefore, the *y*-intercept is 1 and the slope is 2:

**Step 3:** Find the solution. We see that the equations intersect at the point

**(1, 3)**.

**Step 4:** Check the solutions in both equations. We can easily substitute the values of

*x=1*and

*y=3*to check that both equations are true:

$latex x+y=4$

$latex 1+3=4$

$latex 4=4$

$latex 2x-y=-1$

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

$latex -1=-1$

### EXAMPLE 2

Solve the system of equations graphically: $latex \begin{cases}x+2y=7 \\ 3x-y=7 \end{cases}$

##### Solution

**Step 1:** Graph the first equation. We can rewrite the equation in the form $latex y=mx+b$, where

*m*is the slope and

*b*is the

*y*-intercept.

$latex x+2y=7$

$latex y=-\frac{1}{2}x+\frac{7}{2}$

Therefore, the *y*-intercept is 7/2 and the slope is -1/2:

**Step 2:** Graph the second equation. We use the same method as the previous equation:

$latex 3x-y=7$

$latex y=3x-7$

Therefore, the *y*-intercept is -7 and the slope is 3:

**Step 3:** Find the solution. We see that the equations intersect at the point

**(3, 2)**.

**Step 4:** We can easily check the solution that by substituting the values of

*x=3*and

*y=2,*both equations are true:

$latex x+2y=7$

$latex 3+2(2)=7$

$latex 7=7$

$latex 3x-y=7$

$latex 3(3)-2=7$

$latex 7=7$

## Solving systems of equations by substitution

Follow these steps to solve systems of equations using the substitution method:

#### 1. Simplify if possible.

This includes removing parentheses or other grouping signs and combining like terms. If we have fractions, we can multiply by the least common multiple.

#### 2. Solve an equation for one variable.

It doesn’t matter which equation we choose or for which variable we solve. If one of the equations is already solved for a variable, we can use that equation.

#### 3. Substitute the equation from step 2 into the other equation.

Be sure to substitute in the equation you didn’t use in step 2. This will give us an equation with only one variable.

#### 4. Solve the equation from step 2 for the remaining variable.

If you need help with this, you can take a look at our guide on how to solve equations with one variable.

#### 5. Solve for the second variable.

Substitute the value you found in step 4 into either equation and solve for the other variable.

#### 6. Check the solution in both equations.

Substitute the values of the variables into both equations. If both equations are true, the values are the correct solution.

### EXAMPLE 1

Solve the system of equations using the substitution method: $latex \begin{cases}3x-5y=15 \\ y=2x+4 \end{cases}$

##### Solution

**Step 1:** Simplify if possible. In this case, both equations are already simplified.

**Step 2:** Solve an equation for one variable. We can see that the second equation is already solved for the variable

*y*:

$latex y=2x+4$

**Step 3:** Substitute what you got in step 2 into the other equation. We substitute the expression $latex y=2x+4$ into the first equation:

$latex 3x-5y=15$

$latex 3x-5(2x+4)=15$

$latex 3x-10x-20=15$

**Step 4:** Solve for the remaining variable:

$latex 3x-10x-20=15$

$latex -7x=35$

$latex x=-5$

**Step 5:** Solve for the second variable. We substitute $latex x=-5$ into the second equation:

$latex y=2x+4$

$latex y=-10+4$

$latex y=-6$

**Step 6:** Check the solution in both equations.

### EXAMPLE 2

Solve the system of equations using the substitution method: $latex \begin{cases}x-2y=3 \\ 2x-3y=7 \end{cases}$

##### Solution

**Step 1:** Both equations are already simplified.

**Step 2:** We can solve the first equation for

*x*:

$latex x-2y=3$

$latex x=3+2y$

**Step 3:** We substitute the expression $latex x=3+2y$ into the second equation:

$latex 2x-3y=7$

$latex 2(3+2y)-3y=7$

$latex 6+4y-3y=7$

**Step 4:** Solve for the remaining variable:

$latex 6+4y-3y=7$

$latex y=1$

**Step 5:** Solve for the second variable. We substitute $latex y=1$ into the first equation:

$latex x-2y=3$

$latex x-2(1)=3$

$latex x=5$

**Step 6:** Check the solution in both equations.

### Try solving the following practice problems

## Solving systems of equations by elimination

Follow these steps to solve a linear system of equations using the substitution method:

#### 1. Simplify if possible and write the equations in the form A*x*+B*y=*C.

This includes removing parentheses or other grouping signs and combining like terms. If we have fractions, we can multiply by the least common multiple.

#### 2. Multiply one or both equations by a number that will create opposite coefficients in a variable.

We are going to add the equations, and we need one of the variables to be eliminated. For example, if we have 2*x* in one equation and 3*x* in the second, we can multiply the first by -3 and the second by 2, thus obtaining -6 in the first and 6 in the second.

#### 3. Add the equations.

When adding the equations, one of the variables will be eliminated, and we will obtain one equation with a single variable.

#### 4. Solve the equation from step 3 for the remaining variable.

Solve the resulting equation from step 3 for the remaining variable. If you need help with this, you can take a look at our guide on how to solve linear equations with one variable.

#### 5. Solve for the second variable.

Substitute the value you found in step 4 into either equation and solve for the other variable.

#### 6. Check the solution in both equations.

Substitute the values of the unknowns into both equations. If both equations are true, the values are the correct solution.

### EXAMPLE 1

Solve the system of equations using the elimination method: $latex \begin{cases}2x+2y=10 \\ -2x+3y=5 \end{cases}$

##### Solution

**Step 1:** Both equations are already simplified and in the form A

*x*+B

*y=*C.

**Step 2:** We already have opposite coefficients in the variable

*x.*

**Step 3:** Add the equations.

$latex 2x+2y=10$

$latex + \hspace{1cm} -2x+3y=5$

___________________

$latex 5y=15$

**Step 4:** Solve for the remaining variable:

$latex 5y=15$

$latex y=3$

**Step 5:** Solve for the second variable. We substitute $latex y=3$ into the first equation:

$latex 2x+2y=10$

$latex 2x+2(3)=10$

$latex 2x+6=10$

$latex 2x=4$

$latex x=2$

**Step 6:** Check the solution in both equations.

### EXAMPLE 2

Solve the system of equations using the elimination method: $latex \begin{cases}2x=y+3 \\ -x+3y=11 \end{cases}$

##### Solution

**Step 1:** Both equations are already simplified. Now, we write them in the form A

*x*+B

*y=*C:

$latex \begin{cases}2x-y=3 \\ -x+3y=11 \end{cases}$

**Step 2:** We multiply the second equation by 2

*:*

$latex \begin{cases}2x-y=3 \\ -2x+6y=22 \end{cases}$

**Step 3:** Add the equations.

$latex 2x-y=3$

$latex + \hspace{1cm} -2x+6y=22$

___________________

$latex 5y=25$

**Step 4:** Solve for the remaining variable:

$latex 5y=25$

$latex y=5$

**Step 5:** Solve for the second variable. We substitute $latex y=5$ into the first equation:

$latex 2x-y=3$

$latex 2x-5=3$

$latex 2x=8$

$latex x=4$

**Step 6:** Check the solution in both equations.

### Try solving the following practice problems

## See also

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