Problem 58 - Algebra 2

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of each equation. Then graph the equation. $$ 4 x-3 y+8=0 $$

Step-by-Step Solution

Answer

The x-intercept is (-2, 0) and the y-intercept is \((0, \frac{8}{3})\).

Step 1: Identify the Intercepts

To find the intercepts, we need to determine where the line crosses the x-axis and the y-axis. The x-intercept occurs where y = 0, and the y-intercept occurs where x = 0.

Step 2: Solve for the X-Intercept

Set y to 0 in the equation and solve for x to find the x-intercept:\[4x - 3(0) + 8 = 0 \]\[4x + 8 = 0 \]Subtract 8 from both sides:\[4x = -8 \]Divide by 4:\[x = -2 \]So, the x-intercept is at (-2, 0).

Step 3: Solve for the Y-Intercept

Set x to 0 in the equation and solve for y to find the y-intercept:\[4(0) - 3y + 8 = 0 \]\[-3y + 8 = 0 \]Subtract 8 from both sides:\[-3y = -8 \]Divide by -3:\[y = \frac{8}{3} \]So, the y-intercept is at \(\left(0, \frac{8}{3}\right)\).

Step 4: Graph the Equation

To graph the equation, plot the intercepts on a coordinate plane. Plot the x-intercept (-2, 0) and the y-intercept \(\left(0, \frac{8}{3}\right)\). Draw a straight line through these two points to graph the equation \(4x - 3y + 8 = 0\).

Key Concepts

x-intercept

The x-intercept is where a graph crosses the x-axis. It's a point, and at that point, the y-value is zero. So, to find the x-intercept, we simply set y to 0 in our equation and solve for x.
For the equation given, we start by setting y to 0:

• Substitute y with 0 into the equation: \(4x - 3(0) + 8 = 0\).
• This simplifies to \(4x + 8 = 0\).
• Next, subtract 8 from each side: \(4x = -8\).
• Finally, divide by 4: \(x = -2\).

So, the x-intercept is at the point (-2, 0). This means the graph of the line crosses the x-axis at x = -2.

y-intercept

The y-intercept is where the graph crosses the y-axis. At this point, the x-value is zero. To find the y-intercept, we set x to 0 and solve for y. This is a straightforward process, which involves substitution and basic algebraic manipulation.
Using the given equation,

• Set x to 0: \(4(0) - 3y + 8 = 0\).
• This simplifies to \(-3y + 8 = 0\).
• Subtract 8 from both sides: \(-3y = -8\).
• Divide by -3: \(y = \frac{8}{3}\).

The y-intercept is at the point \(\left(0, \frac{8}{3}\right)\). This indicates that the line crosses the y-axis a bit above y = 2.

coordinate plane

The coordinate plane is a two-dimensional space that we use to graph equations and visualize concepts like intercepts. It consists of two axes, the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants.

• X-axis: This is the horizontal line where y = 0. It's used to measure distances left and right.
• Y-axis: This is the vertical line where x = 0. It measures distances up and down.
• Quadrants: Numbered I through IV, they help in identifying the sign of coordinates in each section.

When plotting points on this plane, we use coordinates in the form (x, y). The x-intercept (-2, 0) is located on the x-axis, while the y-intercept \((0, \frac{8}{3})\) is positioned on the y-axis. By plotting these intercepts, we can draw a straight line that represents the given linear equation on the coordinate plane.