Problem 83 - Algebra 1

Question

GRAPHING Write the equation in slope-intercept form, and then graph the equation. Label the \(x\) - and \(y\) -intercepts on the graph. $$4 x+2 y-12=0$$

Step-by-Step Solution

Answer

The slope-intercept form of the equation is \(y = -2x + 6\). The graph of this equation has a \(y\)-intercept at point (0,6) and an \(x\)-intercept at point (3,0).

Step 1: Rewrite to Slope-Intercept Form

To convert to slope-intercept form (\(y = mx+b\)), solve the given equation for \(y\):\n\begin{align*}\n4x + 2y - 12 &= 0 \ 2y &= -4x + 12 \ y &= -2x + 6 \n\end{align*}

Step 2: Identify the Slope and Intercept

From the equation \(y = -2x + 6 \), the slope \(m\) is -2 and the \(y\)-intercept \(b\) is 6.

Step 3: Graph the Equation

Plot the \(y\)-intercept (0,6) first. Then, considering the slope is -2, for every 2 units down on the \(y\)-axis, move 1 unit to the right on the \(x\)-axis. Draw the line through these two points.

Step 4: Identify and Label the \(x\)-intercept

The \(x\)-intercept is the \(x\) value where the line intersects the \(x\)-axis (when \(y = 0\)). Substituting \(y = 0\) into the equation \(y = -2x + 6 \) results in \(x = 3\). Label this point (3,0) on the graph.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing linear equations. Basically, it's a straightforward way to look at a linear equation. The slope-intercept form is written as:
\[ y = mx + b \]
Here, \( m \) represents the slope, which tells us how steep the line is. A positive slope means the line goes up as you move to the right, while a negative slope means it goes down. The \( b \) stands for the y-intercept, which is where the line crosses the y-axis. This form makes it easy to graph because you can quickly pinpoint where the line should be starting on the y-axis and at what angle it should be heading based on the slope.

Y-Intercept

The y-intercept is a specific point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is always the constant \( b \). This is the starting point for plotting your graph. You begin by placing a dot on the y-axis where \( y \) equals this constant. From there, you'll use the slope to determine the direction and steepness of the line.

X-Intercept

The x-intercept is where the line crosses the x-axis, and it's calculated by setting \( y \) to zero and solving for \( x \). This can be an insightful part of the graph because it represents the solution to the equation when the output value \( y \) is zero. To find the x-intercept from the slope-intercept form, you plug in \( y = 0 \) and solve for \( x \), giving a point you can also plot on your graph.

Plotting Graphs in Algebra

To graph a linear equation in algebra, start by writing the equation in the slope-intercept form to easily identify the slope and y-intercept. Begin by plotting the y-intercept on the graph.
Using the slope, determine the direction to move from the y-intercept: move up if the slope is positive, and move down if it's negative. The 'rise over run' aspect of slope tells you the relationship between the horizontal and vertical movements. After determining the x-intercept, you should now have at least two points to draw your line, which represents the solution set to the equation. Properly label the x- and y-intercepts, draw the line through them, and extend it across the grid, making sure your line is straight and accurate.