Lesson 44: Solving Proportions Using Cross Products and Slope of a Line

New Concept

The slope of a line is the ratio of the rise to the run between any two points on the line.

What’s next

Next, you’ll master calculating slope from graphs and points, and also learn to solve proportions using the powerful cross products method.

Property

If two ratios are equal, their cross products are equal. For a proportion $\frac{a}{b} = \frac{c}{d}$, this means $ad = bc$.

Examples

• For $\frac{x}{12} = \frac{4}{6}$, cross products give $6x = 12 \cdot 4$, so $6x = 48$ and $x = 8$.
• For $\frac{y}{2.5} = \frac{4}{10}$, cross products give $10y = 2.5 \cdot 4$, so $10y = 10$ and $y = 1$.
• For $\frac{5}{z} = \frac{2}{8}$, cross products give $5 \cdot 8 = 2z$, so $40 = 2z$ and $z = 20$.

Explanation

Turn a tricky fraction equation into a simple one! The 'criss-cross' shortcut lets you multiply diagonally across the equals sign. This gets rid of the fractions and leaves you with a basic equation, making it super easy to find the missing value. It's like a magic trick for proportions!

Property

The slope of a line is the ratio of the rise to the run between any two points on the line.

Examples

• A line that rises 3 units for every 4 units it runs to the right has a slope of $\frac{3}{4}$.
• A line that goes down 5 units for every 2 units it runs to the right has a slope of $\frac{-5}{2}$.
• A perfectly flat horizontal line has a rise of 0, so its slope is $\frac{0}{5} = 0$.

Explanation

Slope tells you how steep a line is. Think of it like a ski hill! A positive slope means you are going uphill from left to right, while a negative slope means you are going downhill. The 'rise' is how much you go up or down, and the 'run' is how far you go across.

Property

The slope between two points is the average rate of change of one variable relative to the other in that interval.

Examples

• A cyclist travels 30 miles in 2 hours. The average rate of change is $\frac{30 \text{ miles}}{2 \text{ hours}} = 15 \text{ miles/hour}$.
• A phone plan's cost increases by 20 dollars over 4 months. The average rate is $\frac{20 \text{ dollars}}{4 \text{ months}} = 5 \text{ dollars/month}$.
• A plant grows 6 inches in 3 weeks. The average rate of growth is $\frac{6 \text{ inches}}{3 \text{ weeks}} = 2 \text{ inches/week}$.

Explanation

This is just a fancy name for slope in real-world situations! It tells you how fast something is changing on average. For example, it can be miles per hour, dollars per item, or pages read per day. It’s the 'rise' (the change in one thing) divided by the 'run' (the change in another).