Problem 71 - Precalculus Essentials

Question

Use intercepts to graph equation. $$8 x-2 y+12=0$$

Step-by-Step Solution

Answer

The x-intercept is -1.5 and the y-intercept is 6. The line graph of the equation \(8x -2y +12 = 0\) crosses the x-axis at -1.5 and y-axis at 6.

Step 1: Find the x-intercept

To find the x-intercept, we set \(y = 0\). Substituting \(y = 0\) in the equation \(8x -2y +12 = 0\), we get \(8x +12 = 0\). Solving this equation for \(x\), we get \(x = -12/8 = -1.5\). So, the x-intercept is \(-1.5\).

Step 2: Find the y-intercept

To find the y-intercept, we set \(x = 0\). Substituting \(x = 0\) in the equation \(8x -2y +12 = 0\), we get \(-2y +12 = 0\). Solving this equation for \(y\), we get \(y = 12/2 = 6\). So, the y-intercept is \(6\).

Step 3: Graph the equation using intercepts

Now we plot the x-intercept and the y-intercept on the Cartesian coordinate system. The x-intercept is \(-1.5\) and is plotted on the x-axis. The y-intercept is \(6\) and is plotted on the y-axis. Joining these two points with a straight line gives the graph of the equation \(8x -2y +12 = 0\).

Key Concepts

x-intercept

When graphing linear equations, the x-intercept is a key element to find. It's the point where the graph crosses the x-axis. To find the x-intercept, you simply set the value of y to zero in the equation and solve for x. For example, in the equation \(8x - 2y + 12 = 0\), substitute \(y = 0\), resulting in \(8x + 12 = 0\). When you solve for x, you get \(x = -1.5\). So, the x-intercept is \(-1.5\). This means the line crosses the x-axis at this point.

• **x-intercept** is found by setting y to 0.
• It indicates where the line meets the x-axis in the graph.

y-intercept

Another crucial point when graphing is the y-intercept. This is the point where the line crosses the y-axis. To find it, set the value of x to zero and solve for y. Using the equation \(8x - 2y + 12 = 0\), we substitute \(x = 0\) to get \(-2y + 12 = 0\). Solving for y gives \(y = 6\). Thus, the y-intercept is 6.

• **y-intercept** is found by setting x to 0.
• Shows where the line meets the y-axis in the graph.

Knowing both intercepts makes it simple to sketch the graph of a line.

Cartesian coordinate system

The Cartesian coordinate system is the framework used to graph equations like \(8x - 2y + 12 = 0\). It's a grid made of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each point on this grid is identified by a pair of numbers (x, y).

• The **x-axis** runs horizontally and the **y-axis** runs vertically.
• Each point is described by coordinates \((x, y)\).

To graph a line using intercepts, mark the x-intercept on the x-axis and the y-intercept on the y-axis. In our example, plot \(-1.5\) on the x-axis and 6 on the y-axis. Drawing a straight line through these points provides the graph of the given linear equation.