Solve the following system of linear equations by graphing. Graph on your own piece of paper. In your submitted answer, describe what the graph looks like and what this tells you about the solution to the system of linear equations.
y=x+3
−4x+4y=28
(2 points)
5 snettes max
To graph the system of linear equations, we need to plot the points that satisfy both equations and see where they intersect.
Starting with the first equation y = x + 3, we can find several points that satisfy this equation by selecting different values for x and calculating the corresponding values for y. For example, when x = 0, y = 0 + 3 = 3. So, one point on the graph is (0, 3). Similarly, when x = 1, y = 1 + 3 = 4. So, another point is (1, 4).
For the second equation -4x + 4y = 28, let's rewrite it in terms of y to find points on the graph. Divide both sides by 4 to get y = x + 7. Again, we can choose different x values to find corresponding y values. When x = 0, y = 0 + 7 = 7. So, one point on the graph is (0, 7). When x = 1, y = 1 + 7 = 8. So, another point is (1, 8).
Now, let's plot these points on a graph:
------------
y-axis
|
|
| * (0, 7)
| \
| \
| * (0, 3)
| \
| \
| * (1, 8)
|
|
x-axis
------------
By connecting the dots, we can see that the lines intersect at (1, 4).
This tells us that the solution to the system of linear equations is x = 1 and y = 4.
To solve the system of linear equations by graphing, we need to plot the points that satisfy each equation and observe where they intersect.
First, let's graph the equation y = x + 3. This is a linear equation in slope-intercept form (y = mx + b), where the slope (m) is 1 and the y-intercept (b) is 3.
Start by plotting the y-intercept at (0, 3). Then, use the slope (which indicates a ratio of 1 unit increase in x with 1 unit increase in y) to plot more points. For example, moving one unit to the right from the y-intercept gives us the point (1, 4), and moving one unit to the left gives us the point (-1, 2). By connecting these points, we obtain a straight line.
Now, let's graph the equation -4x + 4y = 28. To make it easier to graph, we can rewrite the equation in slope-intercept form by isolating y. Dividing both sides by 4, we get y = x + 7.
The slope of this line is also 1, and the y-intercept is 7. Plotting these points and connecting them, we obtain another straight line.
Now we need to analyze the intersection of the two lines. Looking at the graph, we can see that the lines intersect at the point (5, 8).
This intersection point represents the solution to the system of linear equations. Hence, the solution to the system is x = 5 and y = 8.
In summary, the graph of the system of equations has two lines intersecting at the point (5, 8), indicating that x = 5 and y = 8 is the unique solution to this system.
To solve the system of linear equations by graphing, we need to graph the two equations on a coordinate plane and find the point where they intersect.
Let's start by graphing the first equation, y = x + 3. To do this, we can use the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. In this case, the slope is 1 (the coefficient of x) and the y-intercept is 3.
On your graph paper, plot the point (0, 3) which represents the y-intercept. From that point, use the slope to locate one more point. Since the slope is 1, we can move one unit up and one unit to the right from (0, 3) and mark another point. Connect these two points to draw a straight line.
Next, let's graph the second equation, −4x + 4y = 28. We need to rewrite this equation in slope-intercept form.
-4x + 4y = 28
Rearranging this equation, we get:
4y = 4x + 28
y = x + 7
This equation also has a slope of 1 (coefficient of x) and a y-intercept of 7.
On your graph paper, plot the point (0, 7) which represents the y-intercept. From there, use the slope to locate another point. Since the slope is 1, we can move one unit up and one unit to the right from (0, 7) and mark another point. Connect these two points to draw a straight line.
Now, we can see what the graph looks like. The two lines will intersect at a point. This point represents the solution to the system of equations. In this case, it seems like the lines intersect at the point (4, 7).
The graph of the system of equations shows that there is one solution, which is the point (4, 7). This means that the two equations have a unique point of intersection, and that is the solution to the system.