Question

Line A passes through the point \((0,17)\) on the \(y\)-axis and continues to the right, dropping 3 units for every unit to the right. Line \(B\) is given by \(5 x-3 y=30\). Which line has the larger \(x\)-intercept? $$ \begin{array}{|l|l|} \hline \text { Select... } & \nabla \\ \hline \text { Line A } & \\ \hline \text { Line B } & \\ \hline \end{array} $$

Short answer

Line B has the larger x-intercept (6).

Step-by-step solution

Determine the slope of Line A

Line A drops 3 units for every 1 unit to the right. This means the slope of Line A is \(-3\)

Write the equation of Line A

The line passes through \((0, 17)\), so the y-intercept \(b\) is 17. Using the slope-intercept form \(y = mx + b\), the equation of Line A is \(y = -3x + 17\)

Find the x-intercept of Line A

To find the x-intercept, set \(y = 0\) and solve for \(x\): \ 0 = -3x + 17 \ 3x = 17 \ x = \frac{17}{3} \ The x-intercept of Line A is \(\frac{17}{3}\).

Rewrite the equation of Line B in slope-intercept form

Start with the given equation \(5x - 3y = 30\). Solve for \(y\) to put it in slope-intercept form \(y = mx + b\): \ 5x - 3y = 30 \ -3y = -5x + 30 \ y = \frac{5}{3}x - 10\

Find the x-intercept of Line B

To find the x-intercept, set \(y = 0\) and solve for \(x\): \ 0 = \frac{5}{3}x - 10 \ \frac{5}{3}x = 10 \ x = \frac{10 \cdot 3}{5} = 6\ The x-intercept of Line B is \(6\).

Compare the x-intercepts

Line A has an x-intercept of \(\frac{17}{3} \approx 5.67\) and Line B has an x-intercept of \(6\). So, Line B has the larger x-intercept.

Key concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a convenient way to express lines on a coordinate plane.
It is written as:
(1) \ y = mx + b
Here, m represents the slope of the line, and b signifies the y-intercept, the point where the line crosses the y-axis.
For Line A, which drops 3 units for each unit to the right, the slope m is -3. Using the given point (0, 17), we see that the y-intercept b is 17.
Thus, the equation of Line A becomes:
\( y = -3x + 17 \)
For Line B, the equation is given in standard form: \(5x - 3y = 30\). To convert it to slope-intercept form:

• Subtract 5x from both sides: \(-3y = -5x + 30\)
• Divide by -3: \(y = \frac{5}{3}x - 10\)

Now, Line B is in slope-intercept form as well:
\( y = \frac{5}{3}x - 10 \) This representation helps in quickly identifying the slope and the y-intercept which are essential for graphing.

Solving Linear Equations

Solving linear equations often involves finding intercepts or specific values of x and y that satisfy the equation.
To find the x-intercept, set y to 0 and solve for x. For Line A, given by \(y = -3x + 17\), set y to 0:

• \(0 = -3x + 17\)
• Solving the equation: \(3x = 17 \implies x = \frac{17}{3}\), which is approximately 5.67.

For Line B, the converted equation is \( y = \frac{5}{3}x - 10\):

• Set y to 0: \(0 = \frac{5}{3}x - 10\)
• Solve for x: \(\frac{5}{3}x = 10 \implies x = 6\).

These steps outline the core approach for finding x-intercepts by setting y to zero and isolating x. This process is straightforward and relies on basic algebra.

Comparing Values

Once the intercepts are found, comparing them helps determine which line extends farther along the x-axis.
For Line A, the x-intercept is approximately 5.67. For Line B, the x-intercept is 6.
Comparison is simple:

• \( \frac{17}{3} \approx 5.67 \)
• 6 is clearly larger than 5.67

Therefore, Line B has the larger x-intercept.
Comparing values is often a crucial step in various mathematical problems to determine the greater or lesser quantities.
When dealing with linear equations, intercepts give significant information about where the lines intersect the axes, highlighting their behavior and range on the coordinate plane.