Graph the following linear equations on a single sheet of graph paper. Label each line with its respective equation in the format y = mx + b, where m represents the slope and b represents the y-intercept.
* y=9x+8
* y = -2x+4
* 2y=x-6
* -5y=x+4
Answer (2)
Convert each equation to slope-intercept form ( y = m x + b ) to identify the slope and y-intercept.
Plot the y-intercept for each equation on the y-axis.
Use the slope to find another point on each line and draw the line.
Label each line with its respective equation: y = m x + b .
Explanation
Understanding the Problem We are asked to graph four linear equations on a single sheet of graph paper. Each line must be labeled with its equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. We also need to make sure the graph has appropriate x- and y-axes and includes student information.
Converting to Slope-Intercept Form First, we need to express each equation in slope-intercept form:
y = 9 x + 8 is already in slope-intercept form. The slope is 9 and the y-intercept is 8.
y = − 2 x + 4 is already in slope-intercept form. The slope is -2 and the y-intercept is 4.
2 y = x − 6 . To get this in slope-intercept form, we divide both sides by 2: y = 2 1 x − 3 . The slope is 2 1 and the y-intercept is -3.
− 5 y = x + 4 . To get this in slope-intercept form, we divide both sides by -5: y = − 5 1 x − 5 4 . The slope is − 5 1 and the y-intercept is − 5 4 .
Graphing the Lines Now we will create a coordinate plane and graph each line:
For y = 9 x + 8 , plot the y-intercept at (0, 8). Use the slope of 9 to find another point. For example, go 1 unit to the right and 9 units up to the point (1, 17). Draw the line through these two points.
For y = − 2 x + 4 , plot the y-intercept at (0, 4). Use the slope of -2 to find another point. For example, go 1 unit to the right and 2 units down to the point (1, 2). Draw the line through these two points.
For y = 2 1 x − 3 , plot the y-intercept at (0, -3). Use the slope of 2 1 to find another point. For example, go 2 units to the right and 1 unit up to the point (2, -2). Draw the line through these two points.
For y = − 5 1 x − 5 4 , plot the y-intercept at (0, - 5 4 ). Use the slope of − 5 1 to find another point. For example, go 5 units to the right and 1 unit down to the point (5, - 5 9 ). Draw the line through these two points.
Labeling and Final Touches Finally, label each line with its equation in the form y = m x + b . Also, make sure to include appropriate x- and y-axes and your student information on the graph.
Examples
Understanding linear equations and their graphs is crucial in many real-world applications. For instance, in economics, you can model supply and demand curves using linear equations. The slope represents the rate of change, such as how much the price increases for each additional unit supplied. The y-intercept can represent fixed costs or the starting price. By graphing these equations, economists can visually analyze market trends and predict equilibrium points where supply meets demand. This helps in making informed decisions about production, pricing, and resource allocation.
To graph the linear equations, convert each to slope-intercept form to identify slopes and y-intercepts. Then, plot points and draw lines, labeling each with its respective equation. This process helps visualize and understand linear relationships.
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