Problem 61 - College Algebra with Modeling and Visualization

Question

Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Graph the equation. $$ 6 x-7 y=-42 $$

Step-by-Step Solution

Answer

The x-intercept is (-7, 0), and the y-intercept is (0, 6).

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). The equation becomes: \(6x - 7(0) = -42\). Simplifying, we get \(6x = -42\). Divide both sides by 6 to find \(x = -7\). Hence, the x-intercept is \((-7, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). The equation becomes: \(6(0) - 7y = -42\). Simplifying, we get \(-7y = -42\). Divide both sides by -7 to find \(y = 6\). Hence, the y-intercept is \((0, 6)\).

Step 3: Plot the intercepts and draw the line

On a coordinate plane, plot the points \((-7, 0)\) and \((0, 6)\). These are the x-intercept and y-intercept respectively. Draw a straight line through these two points to represent the equation. This line is the graph of the equation \(6x - 7y = -42\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a graph is a crucial point where the graph crosses the x-axis. To find an x-intercept, we set the value of y to zero, because at any point on the x-axis, the y-coordinate is always zero. After setting y to zero, we only need to solve the equation for x.

• Set y = 0 in the equation.
• Solve the resulting equation for x.
• The solution gives the x-intercept as a coordinate (x, 0).

For example, given the equation \(6x - 7y = -42\), by setting \(y = 0\), it transforms into \(6x = -42\). Solving this gives us \(x = -7\), hence the x-intercept is \((-7, 0)\). This point indicates where the line will touch the x-axis.

Understanding the Y-Intercept

The y-intercept is similar to the x-intercept but occurs where a graph crosses the y-axis. At this point, the x-coordinate is always zero. To find the y-intercept, set x to zero and solve for y.

• Set x = 0 in the equation.
• Simplify the equation to solve for y.
• The solution is the y-intercept as a coordinate (0, y).

Using the same equation, \(6x - 7y = -42\), set \(x = 0\). The equation becomes \(-7y = -42\). Solving it gives \(y = 6\), so the y-intercept is \((0, 6)\). This tells us where the line will intersect the y-axis.

Graphing Linear Equations

Graphing linear equations involves plotting points and drawing a line through them. The x- and y-intercepts provide two key points that help in perfecting this task.

• First, determine the x-intercept and y-intercept.
• Plot these points on a coordinate plane.
• Draw a straight line through the two intercept points.

For the equation \(6x - 7y = -42\), once the intercepts \((-7, 0)\) and \((0, 6)\) are known, place these points on the graph. Drawing a line through them visualizes the equation. This line represents all solutions to the equation, and each point on it satisfies the original equation. Making sure the line is as straight as possible ensures the accuracy of the graph representation.