Problem 65 - Thinking Mathematically

Question

Describe how to find the \(x\)-intercept of a linear equation.

Step-by-Step Solution

Answer

The \(x\)-intercept of a linear equation can be found by setting \(y = 0\) in the equation and then solving for \(x\). This gives the value of \(x\) when the line crosses the \(x\)-axis.

Step 1: Identify the linear equation

Identify the linear equation for which you need to find the \(x\)-intercept. The equation should be in the form \(y = mx + b\) or can be rearranged to this form. The term \(m\) gives you the slope and \(b\) is the y-intercept.

Step 2: Set \(y = 0\)

The \(x\)-intercept is the point where the line crosses the \(x\)-axis, at which \(y = 0\). So, set the \(y\) term in your equation to 0 to solve for the \(x\)-intercept.

Step 3: Solve for \(x\)

Rearrange the equation to solve for \(x\). This may involve subtracting or adding terms and dividing or multiplying by coefficients. The value you get for \(x\) is the \(x\)-intercept of the equation. This is the value of \(x\) when \(y = 0\).

Key Concepts

x-intercept

The x-intercept of a line is where the line crosses the x-axis of a graph. This point occurs where the output value (y) is zero. Finding the x-intercept involves solving the equation of the line when y equals zero. Here's how you can do it:

• First, ensure that you have your equation in the slope-intercept form, which is typically written as \(y = mx + b\). If it isn't in this form, rearrange it so that it becomes \(y = mx + b\).
• Next, replace the y in the equation with 0 because the y-coordinate at the x-intercept is always zero. This will make your equation look like this: \(0 = mx + b\).
• Now, solve for x. This typically involves some basic arithmetic: subtract or add terms as necessary, then divide or multiply through by any remaining coefficients. The solution to this value of x is your x-intercept.

At the x-intercept, you will always notice that the x-coordinate is found but the y-coordinate is zero.

slope-intercept form

The slope-intercept form of a linear equation is a way of writing the equation so you can quickly determine the slope and the y-intercept. This form is expressed as \(y = mx + b\).

• The term \(m\) represents the slope of the line. The slope dictates how steep the line is and the direction it tilts. A positive slope means the line rises from left to right, while a negative slope means the line falls from left to right.
• The term \(b\) is the y-intercept. This is simply where the line crosses the y-axis. Unlike the x-intercept, this is the point where x in the equation is zero.

This form is particularly helpful because it gives you critical information at a glance. You quickly see both the slope and where the line crosses the y-axis.

solving equations

Solving equations is a fundamental skill in mathematics and involves finding the value of variables that make the equation true. With linear equations, like those in slope-intercept form, the process is straightforward. Here’s a simple guide:

• Usually, start by simplifying the equation if necessary. Combine like terms on either side of the equation.
• If the equation is not already in the form you need, rearrange it. For instance, to find the x-intercept, you’ll set \(y = 0\) and solve the equation for \(x\).
• Perform basic algebraic operations: add or subtract terms to isolate the variable, multiply or divide to solve for it. The goal is to have the variable on one side of the equals sign and the number or expression on the other.

The result you reach after applying these operations should define the variable you were solving for, confirming its value in the context of the equation. Solving these equations often involves routine steps, but mastering them is key to tackling more complex mathematical problems later on.