Problem 93 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. $$y=-6+2 x$$

Step-by-Step Solution

Answer

The x-intercept is 3 and the y-intercept is -6.

Step 1: Finding x-intercept

Setting \(y = 0\) in the equation gives \(0 = -6 + 2x\). Solving for \(x\) yields\(x = 3\). Therefore, the x -intercept is 3.

Step 2: Finding y-intercept

Setting \(x = 0\) in the equation gives \(y = -6 + 2 * 0 = -6\). Therefore, the y -intercept is -6.

Key Concepts

Understanding Linear Equations

Linear equations are the building blocks of algebra. They form straight lines when graphed on a coordinate plane. A linear equation is typically written in the form: \(y = mx + b\). Here, \(m\) is the slope, and \(b\) is the y-intercept—the point where the line crosses the y-axis.

In our given problem, the equation is \(y = -6 + 2x\). This is a linear equation. The coefficient of \(x\) (which is 2 in this equation) represents the slope, indicating how steep the line is. A slope of 2 means that for every unit you move horizontally (to the right), the line moves up by 2 units.
This linear equation doesn't explicitly show the \(b\) term, but it can easily be rewritten as \(y = 2x - 6\), helping us identify the y-intercept (\(-6\)).

The main characteristic of linear equations is their constant rate of change, which makes them predictable and easy to plot. Knowing how to identify and interpret the slope and intercepts provides a foundation for solving more complex algebraic problems.

Finding the X-Intercept

The x-intercept is the point where a graph crosses the x-axis. At this point, the value of \(y\) is zero. To find the x-intercept for any equation, you simply set \(y\) to zero and solve for \(x\).

Let's apply this to our problem. We start with the equation: \(y = -6 + 2x\). To find the x-intercept, set \(y = 0\):

• Substitute \(0\) for \(y\): \(0 = -6 + 2x\)
• Solve for \(x\): add 6 to both sides: \(6 = 2x\)
• Divide by 2: \(x = 3\)

This means the x-intercept is at the point \((3, 0)\). This point is where the line crosses the x-axis. Intercepts are important because they provide key information about the graph's position and behavior on a coordinate system.

Locating the Y-Intercept

The y-intercept is where the graph crosses the y-axis. At this point, the value of \(x\) is zero. This makes finding the y-intercept straightforward: set \(x = 0\) in the equation and solve for \(y\).

Here's how it works for our equation \(y = -6 + 2x\):

• Set \(x = 0\): \(y = -6 + 2 \times 0\)
• Simplify: \(y = -6\)

Thus, the y-intercept is at the point \((0, -6)\). This tells you where the line intersects the y-axis. Knowing the x- and y-intercepts allows you to sketch the graph quickly by connecting these points with a straight line. The intercepts also reveal valuable insights into the relationship described by the equation.