Problem 91 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. $$y=2 x+12$$

Step-by-Step Solution

Answer

The x-intercept of the graph is \(-6, 0\) and the y-intercept is \(0, 12\).

Step 1: Find the x-intercept

Set \(y = 0\) in the equation and solve for \(x\). Doing this gives us: \(0=2x + 12\).

Step 2: Solve for x

Solve the equation \(0=2x + 12\) by subtracting 12 from both sides to get \(2x = -12\). Then, divide by 2, yielding \(x = -6\). Therefore, the coordinate of the x-intercept is \(-6, 0\).

Step 3: Find the y-intercept

The y-intercept is where \(x = 0\). Substitute \(x = 0\) into the equation to get \(y = 2(0) + 12\).

Step 4: Solve for y

Solving the equation \(y = 2(0) + 12\), gives \(y = 12\). Therefore, the coordinate of the y-intercept is \(0, 12\).

Key Concepts

Understanding the X-Intercept

The x-intercept is an important concept in the graph of a linear equation. It is the point where the graph of the equation crosses the x-axis. To find the x-intercept, we need to understand that at this point, the value of the y-coordinate is always zero. This is because any point on the x-axis has a y-value of zero.

To calculate the x-intercept for the equation given in the exercise, set y equal to zero and solve for x.

• Start with the equation: \(0 = 2x + 12\).
• Solve by isolating x: subtract 12 from both sides, resulting in \(2x = -12\).
• Finally, divide both sides by 2, giving \(x = -6\).

Thus, the coordinate of the x-intercept for this given equation is \((-6, 0)\). This point tells us where the line representing the equation will touch or intersect with the x-axis, an essential step when graphing the equation.

Understanding the Y-Intercept

The y-intercept is another fundamental aspect of the graph of a linear equation. It is the point where the line crosses the y-axis. To find this point, we set the x-coordinate to zero. This works because any point on the y-axis has an x-value of zero.

To find the y-intercept for the equation in the example, substitute x with zero and solve for y:

• Use the equation: \(y = 2(0) + 12\).
• Simplify to find \(y = 12\).

Hence, the y-intercept for the equation \(y = 2x + 12\) is \((0, 12)\). This point tells us where the line will intersect with the y-axis, which is crucial for sketching the graph.

Exploring the Coordinate System

A coordinate system is a framework used to visually represent points in two dimensions, using a pair of numbers. These numbers are known as coordinates and are typically formatted as \((x, y)\). The x-axis runs horizontally, while the y-axis runs vertically.

Understanding the coordinate system is key when discussing concepts like x-intercepts and y-intercepts. Each intercept is a point where the line of an equation meets one of these axes. In the provided exercise:

• The x-intercept \((-6, 0)\) means the line crosses the x-axis at x = -6 with zero height on the y-axis.
• The y-intercept \((0, 12)\) means the line crosses the y-axis at height 12 with zero width on the x-axis.

The intersection points and plotting them on a grid is how one starts to sketch the equation’s line. Seeing these intercepts clearly on the coordinate plane aids in understanding how linear equations behave visually.