Problem 29 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(2 x-6 y=12\)

Step-by-Step Solution

Answer

The x-intercept is (6, 0) and the y-intercept is (0, -2).

Step 1: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 2x - 6y = 12 \).\\[ 2(0) - 6y = 12 \] \Simplify to get: \\[-6y = 12 \] \Divide both sides by -6: \\[ y = -2 \] \Therefore, the y-intercept is \((0, -2)\).

Step 2: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 2x - 6y = 12 \).\\[ 2x - 6(0) = 12 \] \Simplify to get: \\[ 2x = 12 \] \Divide both sides by 2: \\[ x = 6 \] \Therefore, the x-intercept is \((6, 0)\).

Step 3: Graph the Equation

To graph the equation, plot the x-intercept \((6, 0)\) and the y-intercept \((0, -2)\) on a coordinate plane. Connect these two points with a straight line, extending the line in both directions. This line represents the graph of the equation \(2x - 6y = 12\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear equation is the point where the graph crosses the x-axis. When the graph hits the x-axis, the y-coordinate is always zero. To find this point, we set the variable \( y \) to zero and solve the equation for \( x \).
For our given equation \( 2x - 6y = 12 \), we substitute \( y = 0 \). This simplifies the equation to \( 2x = 12 \).
After dividing both sides by 2, we find \( x = 6 \). Thus, the x-intercept is at (6, 0).

• Set \( y = 0 \) in the equation.
• Solve for \( x \) to find the x-intercept.
• The x-intercept is the point where the graph meets the x-axis.

Finding the x-intercept is a crucial step in sketching the graph of a linear equation, as it helps in determining one of the intersection points.

Understanding the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the x-coordinate is zero. To find the y-intercept, we need to substitute \( x = 0 \) into the equation and solve for \( y \).
Using our example equation, \( 2x - 6y = 12 \), we replace \( x \) with 0, resulting in \( -6y = 12 \).
By dividing each side by -6, we find \( y = -2 \). Thus, the y-intercept of this equation is (0, -2).

• Set \( x = 0 \) in the equation.
• Solve for \( y \) to determine the y-intercept.
• The y-intercept is where the graph intersects the y-axis.

Knowing the y-intercept gives us a fixed point on the graph, simplifying the graphing process.

Graphing Linear Equations Simplified

Graphing a linear equation involves drawing a straight line that represents all solutions of the equation. To graph the equation given, \( 2x - 6y = 12 \), you need both the x-intercept and y-intercept, which we have found as (6, 0) and (0, -2) respectively.
Start by plotting these intercepts on a coordinate grid. The x-intercept (6, 0) lies on the x-axis, and the y-intercept (0, -2) is on the y-axis.
Connect these two points with a ruler to draw a straight line. This line continues infinitely in both directions and represents every solution for the equation.

• Plot the x- and y-intercepts on the graph.
• Use a ruler to draw a straight line through these points.
• Extend the line in both directions to cover the graph.

Graphing linear equations offers a visual way to see all possible solutions and understand the nature of the relationship between the variables in the equation.