Problem 12 - Introductory Algebra for College Students

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Do not graph the equation. \(4 x-5 y=10\)

Step-by-Step Solution

Answer

The \(x\)-intercept of the equation is \(2.5\) and the \(y\)-intercept is \(-2\). So, the intercepts are \((2.5, 0)\) and \((0, -2)\).

Step 1: Find the x-intercept

To find the \(x\)-intercept, set \(y\) equal to zero and solve for \(x\). Given the equation \(4x-5(0)=10\), or \(4x=10\), divide by 4 on both sides to isolate \(x\). So, \(x=\frac{10}{4}=2.5\). Therefore the \(x\)-intercept is \((2.5, 0)\).

Step 2: Find the y-intercept

To find the \(y\)-intercept, set \(x\) to zero and solve for \(y\). From the equation \(4(0)-5y=10\), you get \(-5y=10\). Divide by -5 on both sides to isolate \(y\). So, \(y=\frac{10}{-5} = -2\). Therefore the \(y\)-intercept is \((0, -2)\).

Key Concepts

X-Intercept

The x-intercept of an equation is the point where the graph of the equation crosses the x-axis. This point is significant because it tells us where the value of y is zero. To find this intercept, we set y to 0 in the linear equation and solve for x.
In our example with the equation \(4x - 5y = 10\), we set \(y = 0\) and solve for x. That changes the equation to \(4x = 10\). Dividing each side by 4 to isolate x, we find \(x = \frac{10}{4} = 2.5\). Hence, the x-intercept is the point on the graph \((2.5, 0)\).

• Always: Set y to zero.
• Solve: For x in terms of the other coefficients.
• Remember: The x-intercept is always where y equals zero.

Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. This point occurs when the value of x is zero. To find it, we need to substitute x with 0 in the equation and solve for y.
For our row equation \(4x - 5y = 10\), replacing x with 0 we get \(-5y = 10\). Now, when we divide by -5, we find \(y = \frac{10}{-5} = -2\). Therefore, the y-intercept is represented as \((0, -2)\).

• Important: Set x to zero to find y-intercept.
• Calculate: Solve for y with x set to zero.
• Conclusion: The y-intercept depicts the graph’s vertical crossing.

Linear Equation

Linear equations are fundamental in mathematics and represent a straight line when graphed on a coordinate plane. These equations generally follow the format \(Ax + By = C\), where A, B, and C are constants.
Understanding a linear equation is crucial because it helps in predicting values and understanding relationships between variables. Let's break down the key components:

• Standard form: \(Ax + By = C\) is useful for finding intercepts.
• Slope-Intercept form: Often written as \(y = mx + b\), showcases the slope \(m\) and the y-intercept \(b\).
• Gradient: The slope tells us the steepness of the line and direction of the graph.

Linear equations are omnipresent in various fields such as economics, science, and everyday problem-solving, making their understanding essential.