Problem 66 - College Algebra with Modeling and Visualization

Question

Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Graph the equation. $$ \frac{2}{3} y-x=1 $$

Step-by-Step Solution

Answer

The x-intercept is \((-1, 0)\), and the y-intercept is \((0, \frac{3}{2})\).

Step 1: Find the x-intercept

To find the x-intercept, we need to set \( y = 0 \) in the equation \( \frac{2}{3} y - x = 1 \) and solve for \( x \). Starting with the equation: \[ \frac{2}{3}(0) - x = 1 \] Simplifying gives us: \[ -x = 1 \] Therefore, the x-intercept is: \[ x = -1 \] So the x-intercept is the point \((-1, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, we set \( x = 0 \) in the equation \( \frac{2}{3}y - x = 1 \) and solve for \( y \). Starting with the equation: \[ \frac{2}{3}y - 0 = 1 \] This simplifies to: \[ \frac{2}{3}y = 1 \] To solve for \( y \), multiply both sides by \( \frac{3}{2} \): \[ y = \frac{3}{2} \times 1 \] \[ y = \frac{3}{2} \] So the y-intercept is the point \((0, \frac{3}{2})\).

Step 3: Graph the Equation

Using the intercepts found in Steps 1 and 2, plot the x-intercept \((-1, 0)\) and the y-intercept \((0, \frac{3}{2})\) on the coordinate plane. Draw a straight line through these two points to graph the equation. This line represents all points \((x, y)\) that satisfy the equation \( \frac{2}{3}y - x = 1 \).

Key Concepts

Understanding x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. This means that at the x-intercept, the value of y is always zero. To find this intercept, you substitute zero for y in the equation and solve for x. It's a simple method to identify at which point the line meets the x-axis.

In the example equation \( \frac{2}{3}y - x = 1 \), you set \( y = 0 \) and solve, leading to the equation \( -x = 1 \). Solving for x gives you \( x = -1 \). Therefore, the x-intercept is the point \((-1, 0)\).

This single point tells you where the equation touches the horizontal axis and is crucial for understanding the line's overall behavior. It indicates that the line will pass through the x-coordinate of -1. Because it's clear to graph, this starting point is useful when sketching out the line on graph paper.

Comprehending y-intercept

The y-intercept is where the line crosses the y-axis, meaning the value of x is zero at this point. To find the y-intercept, simply set x to zero and solve the equation for y. This will help you identify where the line will meet the vertical axis.

For our equation \( \frac{2}{3}y - x = 1 \), set \( x = 0 \) and solve \( \frac{2}{3}y = 1 \). By multiplying both sides by \( \frac{3}{2} \), you find \( y = \frac{3}{2} \). Thus, the y-intercept is at the point \((0, \frac{3}{2})\).

Having this point allows you to identify a specific spot on the graph where the line will intersect the y-axis, giving you another key cue for sketching the line accurately on a graph. It's a vital element to determine the direction and slope of the line.

Introduction to graphing lines

Graphing lines involves plotting points and drawing connections to visualize how an equation behaves. After finding both x-intercept and y-intercept, plotting these points on a coordinate plane allows for an easy and accurate line graph.

For the equation \( \frac{2}{3}y - x = 1 \), you have the intercept points \((-1, 0)\) and \((0, \frac{3}{2})\). Mark these points on the graph, then draw a straight line through them. This line represents the set of all possible points \((x, y)\) that satisfy the equation.

The graph provides a visual representation, making it easier to see the relationship between x and y in the equation. It highlights the slope, which describes the steepness and direction of the line, as well as the intercepts, offering a complete view of how the equation behaves across the coordinate plane. Graphing is an excellent tool for interpreting equations in a tangible way, especially in solving or analyzing real-world problems.