Problem 38 - College Algebra with Corequisite Support

Question

For each of the following exercises, find and plot the \(x\) - and \(y\) -intercepts, and graph the straight line based on those two points. $$ 4 x-3 y=12 $$

Step-by-Step Solution

Answer

x-intercept at (3, 0) and y-intercept at (0, -4).

Step 1: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 4x - 3y = 12 \). This gives:\[4x - 3(0) = 12 \4x = 12 \x = \frac{12}{4} = 3\]So, the x-intercept is at \( (3, 0) \).

Step 2: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 4x - 3y = 12 \). This gives:\[4(0) - 3y = 12 \-3y = 12 \y = \frac{12}{-3} = -4\]So, the y-intercept is at \( (0, -4) \).

Step 3: Plot the points and draw the line

Plot the points \( (3, 0) \) and \( (0, -4) \) on the coordinate plane. Draw a straight line through these points. This line represents the equation \( 4x - 3y = 12 \).

Step 4: Verify the intercepts

Re-check the calculations to ensure correctness. For \( x = 3 \), \( 4(3) - 3(0) = 12 \), and for \( y = -4 \), \( 4(0) - 3(-4) = 12 \). Both calculations confirm the intercepts.

Key Concepts

x-intercept

The x-intercept is the point where a line crosses the x-axis. This means that the value of the y-coordinate is zero at this point. In mathematical terms, to find the x-intercept of an equation, you set the value of y to zero and solve for x. For example, in the equation \(4x - 3y = 12\), setting \(y = 0\) gives us the equation \(4x = 12\). Solving for x, we divide both sides by 4, resulting in \(x = 3\). Therefore, the x-intercept is at the point \((3, 0)\). This tells us that the line will cross the x-axis at 3. This step provides a crucial clue about the behavior of the line in relation to the axes.

y-intercept

Just like the x-intercept, the y-intercept is where the line crosses the y-axis. At this intercept, the x-coordinate is zero. Finding the y-intercept involves setting x to zero in your equation and then solving for y. Using our equation \(4x - 3y = 12\), set \(x = 0\) to get \(-3y = 12\). Here, you solve for y by dividing both sides by -3, resulting in \(y = -4\). Thus, the y-intercept is at \((0, -4)\). This is the exact point where the line cuts through the y-axis. Knowing both intercepts is particularly useful as they offer two pivotal points on the graph that help in sketching the complete straight line.

graphing straight lines

Graphing straight lines involves plotting the line on a coordinate plane using known points, usually including the intercepts. Once the x-intercept \((3, 0)\) and the y-intercept \((0, -4)\) are identified, these serve as reference points for drawing the line. By plotting these points on the graph, you can draw a straight line connecting them, representing the equation \(4x - 3y = 12\). This visually shows the relationship between x and y as described by the equation.

• Use a ruler for precision to ensure the line appears straight on the graph.
• Check that the line extends infinitely by drawing arrows on both ends if needed.
• After drawing the line, verify it by plugging additional values of x or y into the equation to see if they fall along the line you drew.

The intercepts serve as convenient points to start with, offering a guarantee that your line is drawn accurately as per the given equation.