Problem 21 - 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

The solutions are (0, -1), (2, 0), and (4, 1). Plot these to graph the line.

Step 1: Rearrange the equation

Let's start by rearranging the given equation in the form of \(y = mx + c\), which makes it easier to graph. The given equation is \(2x - 4y = 4\). To rewrite it in the desired form, solve for \(y\).\[2x - 4y = 4\]Subtract \(2x\) from both sides:\[-4y = -2x + 4\]Now, divide each term by \(-4\):\[y = \frac{1}{2}x - 1\]

Step 2: Choose values for x

Next, we will choose three different values for \(x\) to find corresponding \(y\) values using the equation \(y = \frac{1}{2}x - 1\).Let's select \(x = 0\), \(x = 2\), and \(x = 4\).

Step 3: Calculate y when x = 0

Substitute \(x = 0\) into the equation \(y = \frac{1}{2}x - 1\):\[y = \frac{1}{2}(0) - 1 = -1\]The first solution is \((0, -1)\).

Step 4: Calculate y when x = 2

Substitute \(x = 2\) into the equation \(y = \frac{1}{2}x - 1\):\[y = \frac{1}{2}(2) - 1 = 1 - 1 = 0\]The second solution is \((2, 0)\).

Step 5: Calculate y when x = 4

Substitute \(x = 4\) into the equation \(y = \frac{1}{2}x - 1\):\[y = \frac{1}{2}(4) - 1 = 2 - 1 = 1\]The third solution is \((4, 1)\).

Step 6: Plot the points and draw the graph

Now that we have the three points \((0, -1)\), \((2, 0)\), and \((4, 1)\), we can plot them on a coordinate plane. Once the points are plotted, draw a straight line through them, as this is the graph of the linear equation \(y = \frac{1}{2}x - 1\). The graph will be a straight line that crosses the y-axis at -1 and has a slope of \(\frac{1}{2}\).

Key Concepts

Graphing Lines

Graphing lines on a coordinate plane involves plotting points that satisfy a given equation and connecting them to form a straight line. To graph a linear equation, follow these steps:

• **Convert the equation to slope-intercept form:** This makes it easier to identify the y-intercept and the slope.
• **Find solutions:** Choose simple values for x, and solve for y to get coordinate pairs.
• **Plot the points:** On the grid, put a dot for each point (x, y) found.
• **Draw the line:** Connect these dots with a straight edge, ensuring the line extends to the edges of your graph.

Graphing is a visual way to interpret equations, making it clear how changes in x affect y. You'll see the direction the line goes (up or down) and where it intersects the axes.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a flat surface made up of two intersecting lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants, allowing you to plot points using ordered pairs (x, y). Here’s how it works:

• **Axes Intersection:** The point where the x-axis and y-axis meet is called the origin (0,0).
• **Quadrants:** - Quadrant I: Top right, both x and y are positive. - Quadrant II: Top left, x is negative, y is positive. - Quadrant III: Bottom left, both are negative. - Quadrant IV: Bottom right, x is positive, y is negative.
• **Plotting Points:** When you plot a point (x, y), move right or left from the origin by x units, then move up or down by y units.

Understanding the coordinate plane helps in visualizing how graphs represent equations, providing a straightforward method to see solutions.

Slope-Intercept Form

The slope-intercept form is a common way to write linear equations, represented as: \[ y = mx + c \] In this form, 'm' is the slope of the line, and 'c' is the y-intercept. Here’s what they stand for:

• **Slope (m):** This indicates how steep the line is. It reflects the rate of change in y when x changes. For example, if the slope is \( \frac{1}{2} \), for every increase of 1 in x, y increases by 0.5.
• **Y-Intercept (c):** This is where the line crosses the y-axis, meaning the point where x is 0. From the equation \( y = \frac{1}{2}x - 1 \), the y-intercept is -1.

Using slope-intercept form simplifies graphing and helps in immediately identifying key features of linear equations. It's a powerful tool for quickly sketching graphs, predicting trends, and understanding relationships between variables.