Problem 69 - Algebra for College Students

Question

Find all intercepts for each line. Some of these lines have only one intercept. $$3 x-5 y=15$$

Step-by-Step Solution

Answer

The intercepts are (5, 0) and (0, -3).

Step 1: Find the x-intercept

To find the x-intercept, set y = 0 in the equation and solve for x.Starting with the equation:\[3x - 5y = 15\]Set y = 0:\[3x - 5(0) = 15\]\[3x = 15\]Solve for x:\[x = 5\]So, the x-intercept is (5, 0).

Step 2: Find the y-intercept

To find the y-intercept, set x = 0 in the equation and solve for y.Starting with the equation:\[3x - 5y = 15\]Set x = 0:\[3(0) - 5y = 15\]\[-5y = 15\]Solve for y:\[y = -3\]So, the y-intercept is (0, -3).

Key Concepts

x-intercept

Understanding the x-intercept is key to analyzing the behavior of linear equations. To find the x-intercept of a linear equation, you set the value of y to zero. This is because the x-intercept appears where the line crosses the x-axis, and at all points on the x-axis, the value of y is zero.

y-intercept

The y-intercept reveals where a line crosses the y-axis. To find the y-intercept of a linear equation, you set the value of x to zero. The y-intercept represents the point where the equation passes through the y-axis, and at this point, the value of x is zero.

solving linear equations

Solving linear equations involves manipulating the equation to find the values of x and y that satisfy it. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. To find the intercepts of the equation, set y to zero to solve for the x-intercept and then set x to zero to find the y-intercept.