Problem 48 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(4 x+8 y=12\)

Step-by-Step Solution

Answer

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

Step 1: Find the Y-intercept

To find the y-intercept, set \(x = 0\) in the equation \(4x + 8y = 12\). This simplifies to \(8y = 12\). Solve for \(y\) by dividing both sides by 8, giving \(y = \frac{12}{8} = \frac{3}{2}\). Thus, the y-intercept is \((0, \frac{3}{2})\).

Step 2: Find the X-intercept

To find the x-intercept, set \(y = 0\) in the equation \(4x + 8y = 12\). This simplifies to \(4x = 12\). Solve for \(x\) by dividing both sides by 4, giving \(x = 3\). Thus, the x-intercept is \((3, 0)\).

Step 3: Plot the Intercepts on the Graph

Plot the y-intercept \((0, \frac{3}{2})\) on the graph. Next, plot the x-intercept \((3, 0)\) on the graph. These points are where the line crosses the axes.

Step 4: Draw the Line through the Intercepts

Draw a straight line through the points \((0, \frac{3}{2})\) and \((3, 0)\) on the graph. This line represents the equation \(4x + 8y = 12\).

Key Concepts

Graphing Intercepts

Graphing intercepts is a fundamental technique when graphing linear equations. It helps us easily visualize where a line crosses the x and y axes without needing to calculate many points.
To graph a linear equation using intercepts, first find both the x-intercept and y-intercept. Plot these points on a coordinate grid. Then, simply draw a line through both points. This line represents the solution to the equation.

• Intercepts are where a line crosses the axes.
• Finding intercepts requires basic algebraic manipulation.
• This method is efficient, needing only two points for accurate graphing.

By focusing on the intercepts, we can avoid complex calculations, making graphing more intuitive.

X-Intercept

The x-intercept is the point where a graph crosses the x-axis. It's found by setting the y-value in the equation to zero. This is because the line touches the x-axis when the y-coordinate is zero.
In the equation \(4x + 8y = 12\), find the x-intercept by substituting \(y = 0\):
\[4x + 8(0) = 12\]
This simplifies to \(4x = 12\). Dividing by 4 gives \(x = 3\), so the x-intercept is \((3, 0)\).
Understanding the x-intercept helps us see how far the line reaches along the x-axis before crossing it. This is essential in understanding the equation's relationship to the x-axis.

• Set \(y = 0\) in the equation to find the x-intercept.
• The x-intercept signifies the line's crossing point on the x-axis.
• Calculating the x-intercept offers insight into the equation's behavior.

Y-Intercept

The y-intercept is where the graph of a linear equation crosses the y-axis. It's determined by setting \(x = 0\) in the equation. This is because the line meets the y-axis where the x-value is zero.
Let's examine the equation \(4x + 8y = 12\). To find the y-intercept, substitute \(x = 0\):
\[4(0) + 8y = 12\]
This reduces to \(8y = 12\). Divide both sides by 8 to solve for \(y\), resulting in \(y = \frac{3}{2}\). Thus, the y-intercept is \((0, \frac{3}{2})\).
Knowing the y-intercept is crucial as it gives the starting point of the line on the y-axis. It helps in predicting how the line behaves vertically on the graph.

• Set \(x = 0\) in the equation to find the y-intercept.
• The y-intercept indicates where the line starts on the y-axis.
• It's pivotal for understanding the line's interaction with the y-axis.