Lesson 6-5: Write and Solve Systems of Equations

Property

To solve word problems with two unknown quantities, translate the sentences into a system of linear equations. Follow this problem-solving strategy:
Step 1. Read the problem. Make sure all the words and ideas are understood.
Step 2. Identify what we are looking for.
Step 3. Name what we are looking for. Choose variables to represent those quantities.
Step 4. Translate into a system of equations.
Step 5. Solve the system of equations using good algebra techniques.
Step 6. Check the answer in the problem and make sure it makes sense.
Step 7. Answer the question with a complete sentence.

Examples

• The sum of two numbers is 25. One number is 5 more than the other. Find them. Let the numbers be $x$ and $y$. The system is $\begin{cases} x+y=25 \\ x=y+5 \end{cases}$. Substituting gives $(y+5)+y=25$, so $2y=20$ and $y=10$. Then $x=10+5=15$. The numbers are 10 and 15.

• The perimeter of a rectangle is 40 inches. The length is 4 inches less than twice the width. Find the dimensions. Let $L$ be length and $W$ be width. The system is $\begin{cases} 2L+2W=40 \\ L=2W-4 \end{cases}$. Substituting gives $2(2W-4)+2W=40$, so $6W-8=40$, $6W=48$, and $W=8$. Then $L=2(8)-4=12$. The length is 12 inches and the width is 8 inches.

Property

The most efficient method for solving a system of linear equations depends on the initial form of the given equations:

• Graphing: Best when both equations are in slope-intercept form ($y = mx + b$) or when a visual estimate is needed.
• Substitution: Best when at least one equation has an isolated variable (e.g., x = ... or y = ...) or a variable with a coefficient of 1 or -1.
• Elimination: Best when both equations are in standard form ($Ax + By = C$), especially if a pair of variables has the same or opposite coefficients.

Examples

• Graphing: For the system $y = 2x + 4$ and $y = -x + 1$, graphing is efficient because both equations are in slope-intercept form and can be easily plotted.
• Substitution: For the system $y = 3x - 2$ and $4x + 2y = 16$, substitution is most efficient because y is already isolated in the first equation.
• Elimination: For the system $5x - 2y = 10$ and $3x + 2y = 14$, elimination is most efficient because both equations are in standard form and the y-terms (-2y and 2y) are opposites that will immediately cancel.

Explanation

While any system of linear equations can be solved using graphing, substitution, or elimination, choosing the most efficient method saves time and reduces the chance of calculation errors. By analyzing the initial format of the equations, you can determine the quickest path to the solution.