Lesson 4.1: Solve Systems of Linear Equations with Two Variables

New Concept

This lesson introduces systems of linear equations, where we find the one point $(x, y)$ that solves two equations at once. You'll master three powerful methods—graphing, substitution, and elimination—to find these unique solutions.

What’s next

First, we'll confirm what a solution is. Then, you'll tackle interactive examples for solving systems by graphing, substitution, and elimination.

Property

When two or more linear equations are grouped together, they form a system of linear equations. The solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair $(x, y)$. To determine if an ordered pair is a solution, substitute the values into each equation. If the ordered pair makes both equations true, it is a solution to the system.

Examples

• Is $(1, -3)$ a solution to the system $\begin{cases} 3x + y = 0 \\ x + 2y = -5 \end{cases}$? For the first equation, $3(1) + (-3) = 0$, which is true. For the second, $1 + 2(-3) = -5$, which is true. Yes, it is a solution.

• Is $(2, -2)$ a solution to the system $\begin{cases} x - 3y = -8 \\ -3x - y = 4 \end{cases}$? For the first equation, $2 - 3(-2) = 8$, which is not $-8$. No, it is not a solution.

Property

To solve a system of linear equations by graphing:
Step 1. Graph the first equation.
Step 2. Graph the second equation on the same rectangular coordinate system.
Step 3. Determine whether the lines intersect, are parallel, or are the same line.
Step 4. Identify the solution to the system.

• If the lines intersect, identify the point of intersection.
• If the lines are parallel, the system has no solution.
• If the lines are the same, the system has an infinite number of solutions.

Step 5. Check the solution in both equations.

Examples

• To solve $\begin{cases} y = x + 1 \\ y = -x + 3 \end{cases}$, we graph both lines. They intersect at the point $(1, 2)$. Checking this point in both equations confirms it is the solution.

• To solve $\begin{cases} y = 2x + 1 \\ y = 2x - 3 \end{cases}$, we graph both lines. They have the same slope ($m=2$) but different y-intercepts, so they are parallel. There is no solution to this system.

Property

A consistent system of equations is a system of equations with at least one solution. An inconsistent system of equations is a system of equations with no solution. If two equations are independent, they each have their own set of solutions (intersecting or parallel lines). If two equations are dependent, all the solutions of one equation are also solutions of the other (coincident lines).

Examples

• The system $\begin{cases} y = 3x - 1 \\ y = -2x + 4 \end{cases}$ has different slopes. The lines will intersect at one point. It is a consistent and independent system.

Property

To solve a system of equations by substitution:

• Step 1. Solve one of the equations for either variable.
• Step 2. Substitute the expression from Step 1 into the other equation.
• Step 3. Solve the resulting equation.
• Step 4. Substitute the solution in Step 3 into either of the original equations to find the other variable.
• Step 5. Write the solution as an ordered pair.
• Step 6. Check that the ordered pair is a solution to both original equations.

Examples

• For $\begin{cases} y = 2x - 1 \\ 3x + 2y = 19 \end{cases}$, substitute $2x-1$ for $y$ in the second equation: $3x + 2(2x - 1) = 19$. This simplifies to $7x = 21$, so $x=3$. Then $y = 2(3) - 1 = 5$. The solution is $(3, 5)$.

• For $\begin{cases} x = 3y + 2 \\ 2x - 5y = 6 \end{cases}$, substitute $3y+2$ for $x$ in the second equation: $2(3y + 2) - 5y = 6$. This simplifies to $y + 4 = 6$, so $y=2$. Then $x = 3(2) + 2 = 8$. The solution is $(8, 2)$.

Property

To solve a system of equations by elimination:

• Step 1. Write both equations in standard form. If any coefficients are fractions, clear them.
• Step 2. Make the coefficients of one variable opposites by multiplying one or both equations by appropriate numbers.
• Step 3. Add the equations to eliminate one variable.
• Step 4. Solve for the remaining variable.
• Step 5. Substitute the solution from Step 4 into one of the original equations to solve for the other variable.
• Step 6. Write the solution as an ordered pair and check it.

Examples

• For $\begin{cases} 2x + 3y = 1 \\ 4x - 3y = 17 \end{cases}$, the $y$ coefficients are opposites. Adding the equations gives $6x = 18$, so $x=3$. Substituting into the first equation gives $2(3) + 3y = 1$, so $3y=-5$ and $y=-\frac{5}{3}$. The solution is $(3, -\frac{5}{3})$.

• For $\begin{cases} 3x + 2y = 7 \\ x - 5y = -3 \end{cases}$, multiply the second equation by $-3$ to get $\begin{cases} 3x + 2y = 7 \\ -3x + 15y = 9 \end{cases}$. Adding them gives $17y=16$, so $y=\frac{16}{17}$. Substituting gives $x = \frac{19}{17}$. The solution is $(\frac{19}{17}, \frac{16}{17})$.

Property

Examples

• For $\begin{cases} y = 4x - 9 \\ 2x + 5y = 12 \end{cases}$, the first equation is already solved for $y$. Substitution is most convenient.

• For $\begin{cases} 5x - 3y = 11 \\ 2x + 7y = -5 \end{cases}$, both equations are in standard form with variables aligned. Elimination is most convenient.