Problem 23 - Prealgebra

Question

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

Step-by-Step Solution

Answer

Points: (0,1), (2,0), (4,-1); graph is a straight line with slope -1/2.

Step 1: Equation Rearrangement

Start by rearranging the given equation to solve for one variable in terms of the other. We'll solve for \(y\) in terms of \(x\).The initial equation is: \[ 2x + 4y = 4 \] Subtract \(2x\) from both sides: \[ 4y = -2x + 4 \] Divide every term by 4 to solve for \(y\): \[ y = -\frac{1}{2}x + 1 \] "},{

Step 2: Choosing Values for x and Solving for y

We'll substitute three different values of \(x\) into the linear equation to find corresponding \(y\) values. This helps generate coordinate points for the graph.1. Choose \(x = 0\): \[ y = -\frac{1}{2}(0) + 1 = 1 \] Point: \((0, 1)\)2. Choose \(x = 2\): \[ y = -\frac{1}{2}(2) + 1 = 0 \] Point: \((2, 0)\)3. Choose \(x = 4\): \[ y = -\frac{1}{2}(4) + 1 = -1 \] Point: \((4, -1)\)

Step 3: Plotting the Points

Now plot the points that we have calculated on the graph: - \((0, 1)\)- \((2, 0)\)- \((4, -1)\)Draw a straight line through these points since this is a linear equation.

Step 4: Verifying the Line

To ensure that your line is accurate, check that all the selected points lie on the line graphically. Double-check: - The slope (change in y over change in x) between any two points should be -1/2. - The line crosses the y-axis at 1, as solved in the rearranged equation.

Key Concepts

Graphing Linear Equations

Graphing linear equations is one of the most fundamental concepts in algebra. At its core, a linear equation represents a straight line on a graph. Each linear equation can be written in the form \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
To graph a linear equation, you will need to calculate points that lie on the line. This is typically done by picking different values for \(x\) and calculating the corresponding \(y\) values. Once you have at least two points, you can draw a line through them to graph the equation.

• **Slope (\(m\))**: Determines the steepness of the line. Positive slopes rise, while negative slopes fall.
• **Y-intercept (\(b\))**: The point where the line crosses the y-axis.

These elements together guide the plotting of any linear equation on a graph. Simply plot the points and connect them.

Solving for y

To graph a linear equation like \(2x + 4y = 4\), it’s often easier to first rearrange it to solve for \(y\). This step transforms the equation into a more familiar form: \(y = mx + b\).
In our example, you start with the equation: \(2x + 4y = 4\). You can isolate \(y\) by:

• Subtracting \(2x\) from both sides to get: \(4y = -2x + 4\).
• Dividing every term by 4 to solve for \(y\): \(y = -\frac{1}{2}x + 1\).

Now, \(y\) is expressed directly in terms of \(x\), making it straightforward to determine \(y\) for any given \(x\). This form is useful because it instantly provides the slope and y-intercept. Understanding how to manipulate equations to solve for \(y\) is a key skill in understanding linear functions.

Coordinate Points

Coordinate points are pairs of numbers that show exact locations on a graph. They are represented in the form \((x, y)\), where \(x\) and \(y\) correspond to positions on the x-axis and y-axis, respectively.
When dealing with linear equations, finding coordinate points involves substituting different values of \(x\) into the equation and solving for \(y\). For example, if you have the equation \(y = -\frac{1}{2}x + 1\), you can choose:

• \(x = 0\) to find \(y = 1\), giving the point \((0, 1)\).
• \(x = 2\) to find \(y = 0\), giving the point \((2, 0)\).
• \(x = 4\) to find \(y = -1\), giving the point \((4, -1)\).

These selected points are then plotted on a graph. When connected with a straight line, they represent the linear equation they were derived from. Coordinate points are vital for graphically understanding and visualizing algebraic relationships.