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)
short
To solve the system of linear equations by graphing, we can start by rearranging the equations in slope-intercept form, which is y = mx + b.
For the first equation, y = x + 3, the slope is 1 and the y-intercept is 3.
For the second equation, rearranging it to slope-intercept form, we have:
-4x + 4y = 28
4y = 4x + 28
y = x + 7
Here, the slope is also 1 and the y-intercept is 7.
Now, we can graph these equations on a piece of paper.
The first equation, y = x + 3, has a y-intercept of 3 and a slope of 1. So, starting at the point (0,3), we can plot another point by going up 1 unit and right 1 unit. We can continue this process to find more points.
The second equation, y = x + 7, has a y-intercept of 7 and a slope of 1. So, starting at the point (0,7), we can plot another point by going up 1 unit and right 1 unit. We can continue this process to find more points.
When we plot these points and connect them, we will see two lines intersecting. The point of intersection is the solution to the system of linear equations.
Upon observing the graph, we can see that the lines intersect at the point (-4, -1). This means that (-4, -1) is the solution to the system of linear equations.
To solve the system of linear equations by graphing, you will need to graph each equation on the same coordinate grid and find the point where the lines intersect. This point represents the solution to the system.
First, let's graph the equation y = x + 3:
To graph this equation, you can start by choosing a few values for x and calculating the corresponding values for y. For example, when x = 0, y = 3. When x = 1, y = 4. When x = -1, y = 2. Plot these points on the coordinate grid and draw a straight line through them. This line represents the graph of the equation y = x + 3.
Next, let's graph the equation -4x + 4y = 28:
To graph this equation, you can rewrite it in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. In this case, we need to rearrange the equation to get y by itself.
Starting with -4x + 4y = 28, we can add 4x to both sides to isolate the y term:
4y = 4x + 28
Then, divide both sides by 4 to solve for y:
y = x + 7
Now that the equation is in slope-intercept form, we know that the y-intercept is 7. We can plot this point on the coordinate grid and use the slope (which is 1) to draw the line.
Once you have graphed both equations on your own piece of paper, look at the graph. Notice the point where the two lines intersect. That point represents the solution to the system of linear equations.
If the lines intersect at a single point, it means that there is a unique solution. If the lines are parallel and do not intersect, there is no solution to the system. If the lines overlap each other completely, they are representing the same equation and have infinitely many solutions.
Describe the graph and interpret what it tells you about the solution to the system of linear equations.
To solve the system of linear equations by graphing, let's start with the first equation:
1) y = x + 3
To graph this equation, we can start by putting it into slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
In this case, the slope (m) is 1, and the y-intercept (b) is 3. The y-intercept tells us that the line crosses the y-axis at the point (0, 3). To find additional points to plot, we can use the slope. For every unit increase in x, there is a corresponding unit increase in y. So, starting from the y-intercept, we can move one unit to the right and one unit up to get the point (1, 4). Similarly, moving one unit to the left and one unit down gives us the point (-1, 2).
By connecting these points, we can draw a line that represents the graph of the first equation.
Next, let's move on to the second equation:
2) -4x + 4y = 28
We can rearrange this equation to get it in slope-intercept form (y = mx + b):
4y = 4x + 28
y = x + 7
Similarly to the first equation, the slope is 1 and the y-intercept is 7. This means that the line crosses the y-axis at (0, 7). By using the same process as before, we can find the points (1, 8) and (-1, 6) to plot on the graph.
Now, let's plot these lines on a graph:
The graph of the first equation (y = x + 3) is a straight line that goes through the points (0, 3), (1, 4), and (-1, 2).
The graph of the second equation (y = x + 7) is also a straight line that passes through the points (0, 7), (1, 8), and (-1, 6).
By graphing these two equations on the same coordinate plane, we can see that the two lines intersect at the point (2, 5).
Therefore, the solution to the system of linear equations is x = 2 and y = 5.
Note: It is important to remember that the accuracy of your graph depends on how accurately you plot the points and connect them to form the lines.