Problem 35 - Prealgebra

Question

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

Step-by-Step Solution

Answer

Graph the line using the slope-intercept form: plot (0, 2) and (5, 0), then draw the line through these points.

Step 1: Transform the Equation to Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by solving the given equation for \( y \). The given equation is \( 2x + 5y = 10 \). To isolate \( y \), subtract \( 2x \) from both sides to get \( 5y = -2x + 10 \). Next, divide all terms by 5 to solve for \( y \), resulting in \( y = -\frac{2}{5}x + 2 \).

Step 2: Identify the Slope and Y-Intercept

From the equation \( y = -\frac{2}{5}x + 2 \), identify the slope and y-intercept. The slope \( m \) is \(-\frac{2}{5}\), and the y-intercept \( b \) is 2. This means the line crosses the y-axis at (0, 2) and slopes downward, as indicated by the negative slope.

Step 3: Plot the Y-Intercept

On the graph, locate the y-axis and plot the point (0, 2), which is the y-intercept. This is your starting point for drawing the line.

Step 4: Use the Slope to Find Another Point

Starting from the y-intercept (0, 2), use the slope to find another point. The slope is \(-\frac{2}{5}\), meaning you move two units down and five units to the right. From (0, 2) go down to (0, 0) and over to (5, 0) to plot the second point.

Step 5: Draw the Line

Using a ruler, draw a straight line through the points (0, 2) and (5, 0), extending it in both directions. This line represents the graph of the equation \(2x + 5y = 10\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a convenient way to write a linear equation so you can easily see key information about the line, including the slope and where it intersects the y-axis.
The general format of a linear equation in slope-intercept form is:

• \(y = mx + b\),where \(m\) is the slope,
• \(b\) is the y-intercept.

This form allows you to quickly determine the direction and steepness of the line.
The slope, \(m\), tells us how much \(y\) changes for a unit change in \(x\).
A positive slope means the line rises from left to right while a negative slope means the line falls.
The y-intercept, \(b\), indicates where the line crosses the y-axis.
In the exercise, the equation is transformed into the slope-intercept form: \(y = -\frac{2}{5}x + 2\),showing the slope \(-\frac{2}{5}\) and y-intercept 2 clearly.
This makes it easier to graph the equation.

Slope and Y-Intercept

The slope and y-intercept are essential components in understanding and graphing linear equations.

• The slope (\(m\)) is a measure of the line's steepness. It describes the rate of change or how much \(y\) increases or decreases as \(x\) increases.
• For our equation, \(y = -\frac{2}{5}x + 2\), the slope is \(-\frac{2}{5}\). This negative slope tells us the line slopes downwards as we move from left to right.

The fraction \(-\frac{2}{5}\) invites a more granular inspection, indicating that:

• For every 5 units you move right on the \(x\)-axis,
• The corresponding \(y\)-value decreases by 2 units.

The y-intercept (\(b\)) shows where the line crosses the y-axis. Here, \(b = 2\), meaning the line will pass through the point (0, 2).
Having a clear y-intercept helps you start the graph at a fixed point, ensuring that the rest of the line will follow suit from this initial anchor.

Plotting Points

Once you've identified the slope and y-intercept from the equation, plotting these on a graph can make the abstract concept more tangible.

• Start with the y-intercept: For our equation, \(y = -\frac{2}{5}x + 2\), locate \( (0, 2) \) on the graph and mark it with a point.
• Use the slope to find another point: Begin at the y-intercept, move according to the slope \(-\frac{2}{5}\). Move 2 units down and 5 units to the right.
• Plot this second point, which brings you to \( (5, 0) \).

Connecting these points with a straight line visualizes the equation.
Make sure to extend the line in both directions, representing all solutions of the linear equation.
Seeing the line on a graph helps comprehend how all solutions lie perfectly on this straight path.