My book is showing the solution for graphing a system of linear equations
y > 2x - 7
y is greater than or equal to -3x + 7
I see the way they graphed it and both lines intersect at one point. Did the just find -7 and +7 on the line in order to graph or did they solve for y and find x in order to fingure out how to graph these two equations. I'm lost.
To graph a system of linear equations, there are a few different approaches you can take. Let's go through a step-by-step process to understand how to graph the given system of linear equations:
1. Start by identifying the slope-intercept form of each equation, which is in the form y = mx + b, where m represents the slope and b represents the y-intercept.
For the first equation, y > 2x - 7, we can see that the slope is 2 and the y-intercept is -7.
For the second equation, y ≥ -3x + 7, the slope is -3 and the y-intercept is 7.
2. Plot the y-intercepts on the graph. For the first equation, the y-intercept is -7, so mark a point at (0,-7). For the second equation, the y-intercept is 7, so mark a point at (0,7).
3. Determine the slope of each line. Since the slope is a ratio of the change in y divided by the change in x, you can find two additional points on each line using the slope.
For the first equation, the slope is 2. You can choose any x-value and find the corresponding y-value by substituting it into the equation. For example, if you choose x = 0, you find that y = 2(0) - 7 = -7, giving you the point (0, -7). Another point can be found by choosing x = 1, which results in y = 2(1) - 7 = -5, giving you the point (1, -5).
For the second equation, the slope is -3. You can again choose any x-value and find the corresponding y-value by substituting it into the equation. For example, if you choose x = 0, you find that y = -3(0) + 7 = 7, giving you the point (0, 7). Another point can be found by choosing x = 1, which results in y = -3(1) + 7 = 4, giving you the point (1, 4).
4. Plot the additional points found in the previous step on the graph.
5. Draw a line through the points for each equation. For the first equation, connect the points (0, -7) and (1, -5) with a line. For the second equation, connect the points (0, 7) and (1, 4) with a line.
6. Finally, shade the region on the graph that satisfies both inequalities. In this case, y > 2x - 7 represents the region above the line, and y ≥ -3x + 7 represents the region at or above the line. The shaded area where both inequalities are true is the solution to the system of linear equations.
To summarize, in order to graph the system of linear equations, you need to find the slope and y-intercept of each equation, plot the y-intercept, find additional points using the slope, plot those additional points, and then draw lines connecting the points. Remember to shade the region that satisfies both inequalities to find the solution.
I hope this explanation helps clarify the graphing process for you!