Problem 36 - Prealgebra

Question

The problems below review material we covered in Section 4.9 Graph each equation. $$5 x+2 y=10$$

Step-by-Step Solution

Answer

The graph is a straight line with slope \(-\frac{5}{2}\) and y-intercept at (0, 5).

Step 1: Convert to Slope-Intercept Form

Start by rewriting the given equation in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The original equation is \( 5x + 2y = 10 \). We solve for \( y \) by subtracting \( 5x \) from both sides of the equation to get: \( 2y = -5x + 10 \). Then, divide every term by 2: \( y = -\frac{5}{2}x + 5 \).

Step 2: Identify the Slope and Y-Intercept

From the slope-intercept form \( y = -\frac{5}{2}x + 5 \), identify the slope \( m = -\frac{5}{2} \) and the y-intercept \( b = 5 \). The y-intercept indicates where the line crosses the y-axis, which is at the point (0, 5).

Step 3: Plot the Y-Intercept

On the Cartesian plane, locate the point (0, 5) and plot it on the y-axis. This is the starting point for graphing the line.

Step 4: Use the Slope to Plot a Second Point

The slope \( -\frac{5}{2} \) means that for every move of 2 units to the right along the x-axis, the line goes down 5 units. From the point (0, 5), move 2 units to the right to (2, 3), then move 5 units down to (2, 0). Plot this new point.

Step 5: Draw the Line

Using a ruler, draw a straight line through the two points plotted: (0, 5) and (2, 0). Extend the line in both directions beyond these points to complete the graph of the equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations to make graphing simple. The general formula is given by \( y = mx + b \). In this formula, \( m \) stands for the slope, while \( b \) is the y-intercept. This form is particularly handy as it directly shows the line's slope and the point where it crosses the y-axis.

Converting any linear equation to this form involves isolating \( y \). For example, consider the equation \( 5x + 2y = 10 \). By manipulating the equation to solve for \( y \) (subtract \( 5x \) from both sides and then divide by 2), it rewrites to \( y = -\frac{5}{2}x + 5 \). This is now in the slope-intercept form, revealing both slope and y-intercept directly.

• The form helps identify characteristics of the line quickly.
• Efficient for graphing by providing a clear starting point (the y-intercept).
• Displays the rate of change (slope) at a glance.

Slope

The slope of a line measures its steepness and direction. In the slope-intercept form, \( m \) is the slope. For the equation \( y = -\frac{5}{2}x + 5 \), the slope \( m = -\frac{5}{2} \). This tells us how the y-coordinate changes with respect to the x-coordinate.

The slope can be thought of as "rise over run", indicating how much the line rises or falls as you move rightward on the graph.

• A positive slope means the line ascends as it moves from left to right.
• A negative slope, like \( -\frac{5}{2} \), means the line descends.
• If the slope is zero, the line is horizontal.

Understanding slope is essential for determining how the line behaves. In practical terms, with a slope of \( -\frac{5}{2} \), for every 2 units moved to the right, you move down 5 units, identifying the steep descent of the line.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis. You can find this intercept in the slope-intercept form by looking at the \( b \) term. For instance, in the equation \( y = -\frac{5}{2}x + 5 \), the y-intercept \( b \) is 5. This means the line crosses the y-axis at (0, 5).

The y-intercept gives a reliable starting point for graphing. It is where the line will intersect the y-axis, and from this point, you can use the slope to determine the direction and steepness of the line.

• The y-intercept is easy to locate and plot on a graph.
• It provides an essential reference point for drawing the entire line.
• Understanding its position is essential for interpreting the graph correctly.

By knowing the y-intercept, you know exactly where to start plotting on the graph before using the slope to find other points on the line.