Lesson 4: Solve Applications with Systems of Equations

New Concept

Master word problems by translating real-world scenarios into systems of linear equations. You'll learn to model situations involving geometry, uniform motion, and direct translations, then solve for the unknown quantities.

What’s next

Now, let's put this into practice. You'll tackle interactive examples and a series of practice cards to build your problem-solving skills.

Property

To solve word problems with two unknown quantities, translate the sentences into a system of linear equations. Follow this problem-solving strategy:

1. Read the problem to understand it.
2. Identify what you need to find.
3. Name the unknowns with variables.
4. Translate the words into a system of equations.
5. Solve the system using a convenient algebraic method.
6. Check that the solution makes sense in the context of the problem.
7. Answer the question with a complete sentence.

Examples

• The sum of two numbers is 50. One number is 10 more than the other. Let $x$ and $y$ be the numbers. The system is $\begin{cases} x + y = 50 \\ x = y + 10 \end{cases}$.

• A company has 85 employees. The number of men is five more than twice the number of women. Let $m$ be men and $w$ be women. The system is $\begin{cases} m + w = 85 \\ m = 2w + 5 \end{cases}$.

Property

• Two angles are complementary if the sum of the measures of their angles is $90$ degrees.
• Two angles are supplementary if the sum of the measures of their angles is $180$ degrees.

Examples

• The difference of two complementary angles is 16 degrees. Let the angles be $x$ and $y$. The system is $\begin{cases} x + y = 90 \\ x - y = 16 \end{cases}$. The angles are $53^\circ$ and $37^\circ$.

• Two angles are supplementary. The larger angle is 20 degrees more than the smaller one. Let the angles be $x$ and $y$. The system is $\begin{cases} x + y = 180 \\ y = x + 20 \end{cases}$. The angles are $80^\circ$ and $100^\circ$.

Property

Perimeter problems use the total distance around a shape to form an equation. For a rectangle with length $L$ and width $W$, the perimeter is $P = 2L + 2W$. Some problems may involve fencing only some sides, such as $L + 2W$ for a yard against a house.

Examples

• The perimeter of a rectangular park is 100 meters. The length is 10 meters more than the width. The system is $\begin{cases} 2L + 2W = 100 \\ L = W + 10 \end{cases}$. The dimensions are $L=30$ m and $W=20$ m.

• A fence is built around three sides of a rectangular yard adjacent to a house, using 140 feet of fencing. The length is 20 feet more than twice the width. The system is $\begin{cases} L + 2W = 140 \\ L = 2W + 20 \end{cases}$. The dimensions are $L=68$ ft and $W=36$ ft.

Property

Uniform motion problems use the formula Distance = Rate × Time ($D=rt$). In 'catch-up' scenarios, two objects start at the same point and travel the same route, so their distances are equal when one catches the other. The system of equations sets their distances equal: $D_1 = D_2$.

Examples

• Ken leaves driving 55 mph. An hour later, his sister follows at 65 mph. How long until she catches him? Let $t$ be Ken's time. The equation is $55t = 65(t-1)$. It takes his sister 5.5 hours.

• A biker starts at 12 mph. A scooter starts 30 minutes ($\frac{1}{2}$ hour) later at 20 mph. How long until the scooter catches up? Let $t$ be the biker's time. The equation is $12t = 20(t - 0.5)$. It takes the scooter 45 minutes.

Property

For motion with wind or a current, speeds combine. Let $r$ be the object's speed in still conditions and $c$ be the current/wind speed.

• Speed with tailwind/downstream: $r+c$
• Speed with headwind/upstream: $r-c$

Use the formula $D=rt$ to create an equation for each leg of the journey.

Examples

• A boat travels 45 miles downstream in 3 hours and returns upstream in 5 hours. Let $b$ be boat speed and $c$ be current. The system is $\begin{cases} 3(b+c)=45 \\ 5(b-c)=45 \end{cases}$. Boat speed is 12 mph, current is 3 mph.

• A plane flies 1500 miles in 3 hours with a tailwind, and the return trip against the wind takes 5 hours. Let $p$ be plane speed and $w$ be wind. The system is $\begin{cases} 3(p+w)=1500 \\ 5(p-w)=1500 \end{cases}$. Plane speed is 400 mph, wind is 100 mph.