2003 AMC 10B Problem 11

P11-15~2 minPrint

Coordinate Geometry Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

A line with slope $3$ intersects a line with slope $5$ at point $(10,15)$ . What is the distance between the $x$ -intercepts of these two lines?

Show solution without answering

Key insight

Both lines drop 15 units to reach the x-axis: slope 3 needs 5 steps left, slope 5 needs 3 steps left. The crossings are 5 and 7.

Solution

An $x$ -intercept is where a line reaches height $0$ . Both lines start at the point $(10, 15)$ , which is at height $15$ , so each has to drop $15$ units.
Slope $3$ means: every $1$ step to the right, the line goes up $3$ . Going left, it goes down $3$ per step. To drop $15$ it needs $15 \div 3 = 5$ steps left. Its $x$ -intercept is $10 - 5 = 5$ .
Slope $5$ means it drops $5$ per step left. To drop $15$ it needs $15 \div 5 = 3$ steps. Its $x$ -intercept is $10 - 3 = 7$ .
The distance between the intercepts is $7 - 5 = 2$ .
Check with equations: the lines are $y = 3x - 15$ and $y = 5x - 35$ . Putting $y = 0$ gives $x = 5$ and $x = 7$ .

The answer is $\boxed{\textbf{(A)}\ 2}$ .

Why this works

Slope is "up divided by across," so if you know how far up or down you must go, the distance across is that amount divided by the slope. The steeper line reaches the axis after fewer steps, which is why its intercept is closer to $x = 10$ . This avoids writing equations at all.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Both lines pass through $(10, 15)$ and must descend $15$ units to reach the $x$ -axis.

On the line of slope $3$ , each unit of horizontal movement changes $y$ by $3$ , so descending $15$ requires moving $\frac{15}{3} = 5$ units to the left. Its $x$ -intercept is $10 - 5 = 5$ .

On the line of slope $5$ , descending $15$ requires $\frac{15}{5} = 3$ units to the left, so its $x$ -intercept is $10 - 3 = 7$ .

The distance between the intercepts is $7 - 5 = 2$ .

(Equivalently, the lines are $y = 3x - 15$ and $y = 5x - 35$ ; setting $y = 0$ gives $x = 5$ and $x = 7$ .)

The answer is $\boxed{\textbf{(A)}\ 2}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 17: Analytic Geometry
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 8: Graphing Lines
Introduction to Algebra, Chapter 7: Proportion (§7.4)