5.4 Solve Applications with Systems of Equations

Constructing and Applying Systems of Equations

When a word problem involves two unknown quantities, a single equation usually isn't enough. Systems of equations let you set up two equations with two unknowns, then solve them together. The key skill here is translating English sentences into algebra.

Translating Word Problems into Systems

Every word problem follows the same basic setup process:

1. Read the full problem before writing anything. Figure out what you're being asked to find.
2. Assign variables to the unknown quantities. Be specific: write "let $x$ = the number of adult tickets" rather than just "let $x$ = adults."
3. Find two relationships between the unknowns. Each relationship becomes one equation.
4. Solve the system using substitution or elimination.
5. Check your answer against the original problem (not just the equations). Make sure it actually answers the question.

Watch for key phrases that signal specific operations:

• "the sum of" → addition
• "the difference between" → subtraction
• "twice as much as" → multiply by 2
• "total" or "combined" → addition
• "more than" → addition
• "less than" → subtraction

You need the same number of equations as unknowns. Two unknowns require two equations. If you can only find one equation, re-read the problem for a relationship you missed.

Real-World Applications

Several common problem types show up repeatedly. Recognizing the pattern makes setup faster.

Business/money problems usually give you two pieces of information: a total count and a total value. For example: "A store sold 40 items. Small items cost $5 and large items cost $12. Total revenue was $355. How many of each were sold?"

• Let $x$ = number of small items, $y$ = number of large items
• Equation 1 (total count): $x + y = 40$
• Equation 2 (total value): $5x + 12y = 355$

Mixture problems work the same way. You'll have a total amount of mixture and a total amount of some ingredient (like acid concentration or salt content).

Investment problems involve principal amounts and interest. If someone invests a total of $10,000 in two accounts at different rates, you get one equation for the total principal and one for the total interest earned.

In every case, solve the system, then state your answer in words that match the original question.

Geometric Problem-Solving

Geometry problems give you relationships between measurements (side lengths, angles) and ask you to find the actual values. The geometric formulas become your equations.

Perimeter example: "A rectangle's length is 3 more than twice its width. The perimeter is 54 cm. Find the dimensions."

1. Let $l$ = length, $w$ = width
2. Relationship between dimensions: $l = 2w + 3$
3. Perimeter formula: $2l + 2w = 54$
4. Substitute the first equation into the second: $2(2w + 3) + 2w = 54$
5. Solve: $4w + 6 + 2w = 54$, so $6w = 48$, giving $w = 8$ and $l = 19$
6. Check: $2(19) + 2(8) = 38 + 16 = 54$ ✓

Common geometric relationships you'll use:

• Perimeter of a rectangle: $2l + 2w = P$
• Complementary angles: two angles that add to $90°$
• Supplementary angles: two angles that add to $180°$
• Triangle angle sum: all three angles add to $180°$

Uniform Motion Problems

These problems are built on one formula: $d = rt$ (distance equals rate times time). They typically involve two objects moving, and you set up a $d = rt$ equation for each one.

Common setups:

• Traveling in opposite directions: The distances add up to the total distance apart.
• Traveling in the same direction: The distances are equal when one catches the other.
• Round trips: Same distance going and coming back, but different rates or times.

Example: "Two cars leave the same point traveling in opposite directions. One goes 55 mph and the other goes 65 mph. After how many hours are they 420 miles apart?"

1. Both cars travel the same amount of time. Let $t$ = time in hours.
2. Car 1 distance: $55t$. Car 2 distance: $65t$.
3. Their distances add up: $55t + 65t = 420$
4. Solve: $120t = 420$, so $t = 3.5$ hours.

When rates are unknown, you'll need two equations. For instance, a boat traveling with and against a current gives you: rate downstream = $b + c$ and rate upstream = $b - c$, where $b$ is the boat's speed in still water and $c$ is the current speed.

Additional Techniques

A few strategies help when problems get more complex:

• Variable isolation: If one equation is already solved for a variable (like $y = 3x + 2$), substitution is usually the fastest method.
• Graphical interpretation: Each equation represents a line. The solution to the system is the point where the lines intersect. This can help you estimate or verify answers.
• Checking for reasonableness: Negative values for time, distance, or number of items usually signal an error. Always ask whether your answer makes sense in context.