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 (1)
Rewrite each equation in slope-intercept form: y = m x + b .
Identify the slope m and y-intercept b for each equation.
Plot each line using its slope and y-intercept.
Label each line with its corresponding equation in the format y = m x + b .
Explanation
Understanding the Problem We are asked to graph four linear equations on a single sheet of graph paper. We need to rewrite each equation in slope-intercept form ( y = m x + b ), identify the slope ( m ) and y-intercept ( b ) for each, and then graph the lines. Finally, we label each line with its equation.
Rewriting Equations in Slope-Intercept Form Let's rewrite each equation in slope-intercept form:
y = 9 x + 8 is already in slope-intercept form. Here, m = 9 and b = 8 .
y = − 2 x + 4 is already in slope-intercept form. Here, m = − 2 and b = 4 .
2 y = x − 6 . Divide both sides by 2 to get y = 2 1 x − 3 . Here, m = 2 1 and b = − 3 .
− 5 y = x + 4 . Divide both sides by -5 to get y = − 5 1 x − 5 4 . Here, m = − 5 1 and b = − 5 4 .
Identifying Slopes and Y-Intercepts Now we have the slope and y-intercept for each equation:
y = 9 x + 8 : m = 9 , b = 8
y = − 2 x + 4 : m = − 2 , b = 4
y = 2 1 x − 3 : m = 2 1 , b = − 3
y = − 5 1 x − 5 4 : m = − 5 1 , b = − 5 4
Plotting the Lines To graph these lines, we'll plot the y-intercept and use the slope to find another point. For example, for y = 9 x + 8 , we start at (0, 8). Since the slope is 9, we can go 1 unit to the right and 9 units up to find another point (1, 17). However, since we are graphing all lines on one graph, it is important to choose a scale that allows us to plot all lines.
For y = − 2 x + 4 , we start at (0, 4). Since the slope is -2, we can go 1 unit to the right and 2 units down to find another point (1, 2).
For y = 2 1 x − 3 , we start at (0, -3). Since the slope is 2 1 , we can go 2 units to the right and 1 unit up to find another point (2, -2).
For y = − 5 1 x − 5 4 , we start at (0, - 5 4 ). Since the slope is - 5 1 , we can go 5 units to the right and 1 unit down to find another point (5, - 5 9 ).
Labeling and Finalizing the Graph After plotting the lines, make sure to label each line with its equation in the format y = m x + b . Also, include appropriate x- and y-axes labels and your student information on the graph.
Examples
Understanding linear equations and their graphs is crucial in many real-world applications. For instance, in business, you might use a linear equation to model the cost of producing a certain number of items, where the slope represents the variable cost per item and the y-intercept represents the fixed costs. Graphing these equations allows you to visualize the relationship between production volume and total cost, helping you make informed decisions about pricing and production levels. Similarly, in physics, you can use linear equations to describe the motion of an object at a constant velocity, where the slope represents the velocity and the y-intercept represents the initial position.
