Problem 22 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$4 x+2 y=4$$

Step-by-Step Solution

Answer

The solutions are (0, 2), (1, 0), and (2, -2). Plot these on a graph and connect them with a line.

Step 1: Rewrite the Equation in Slope-Intercept Form

To make it easier to find points to plot, rewrite the equation in the form of \(y = mx + b\). Start by isolating \(y\) in the equation \(4x + 2y = 4\). Subtract \(4x\) from both sides to get \(2y = -4x + 4\). Next, divide each term by 2 to solve for \(y\), resulting in \(y = -2x + 2\).

Step 2: Choose Values for "x"

Select three different values for \(x\) to find corresponding \(y\) values. Commonly chosen values are \(x = 0\), \(x = 1\), and \(x = 2\).

Step 3: Calculate "y" for Each Chosen "x" Value

Use the equation \(y = -2x + 2\) to calculate \(y\) for each chosen \(x\). For \(x = 0\), \(y = -2(0) + 2 = 2\). For \(x = 1\), \(y = -2(1) + 2 = 0\). For \(x = 2\), \(y = -2(2) + 2 = -2\).

Step 4: Plot the Points on the Graph

With the points calculated, plot them on a graph: (0, 2), (1, 0), and (2, -2). These points should align in a straight line because the equation represents a linear function.

Step 5: Draw the Graph

Connect the points with a straight line to represent the graph of the linear equation \(4x + 2y = 4\). Extend the line in both directions and include arrows to indicate it continues indefinitely.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so it is very easy to understand and use. In this form, the equation of the line looks like this: \[ y = mx + b \]This might seem a bit abstract, so let's break it down:

• \( y \) is the value on the y-axis that you are solving for, it changes depending on the x value.
• \( m \) represents the slope of the line. The slope is the steepness or tilt of the line. It's essentially how much \( y \) increases or decreases as \( x \) increases.
• \( x \) is the value on the x-axis, the independent variable that you choose.
• \( b \) is the y-intercept. This is the point where the line crosses the y-axis. It's the value of \( y \) when \( x \) is zero.

To rewrite an equation in this form, you need to isolate \( y \), which allows you to read the slope \( m \) and y-intercept \( b \) straight from the equation. For example, the equation \( 4x + 2y = 4 \) becomes \( y = -2x + 2 \) after simplifying, with a slope of \(-2\) and y-intercept at 2.

Graphing Linear Equations

Graphing linear equations involves visualizing the solutions on a 2D grid. The slope-intercept form makes graphing straightforward because it directly shows you where the line crosses the y-axis and its slope.
The graph of a linear equation is always a straight line. To graph an equation like \( y = -2x + 2 \), start by identifying the y-intercept, which here is 2. Plot this point on the y-axis.
Next, use the slope

• A slope of \(-2\) means for every step to the right (positive x direction), the line moves two steps down.
• From the y-intercept (0,2), move to the right one unit and down two units to plot the next point (1,0).

Continue using the slope to find additional points. Once you've plotted a few points, draw a line through them. Extend the line in both directions, adding arrows at each end to show that it continues indefinitely.

Solving for y

Sometimes, equations are given in forms that aren't immediately useful for graphing directly. That's where solving for y becomes extremely handy.
To solve for y, your goal is to rearrange the equation so that \( y \) is by itself on one side of the equation. Consider the equation \( 4x + 2y = 4 \).

• Start by isolating terms with \( y \). Subtract \( 4x \) from both sides to rewrite it as \( 2y = -4x + 4 \).
• To get \( y \) by itself, divide every term by 2. This gives you \( y = -2x + 2 \).

By solving for \( y \), you uncover both the slope and y-intercept, turning the equation into a handy format for drawing graphs or analyzing the line's behavior.