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)
To solve the given system of linear equations by graphing, we need to graph each equation on a coordinate plane and find the point of intersection. The point of intersection will be the solution to the system.
First, let's graph the equation y = x + 3. To do this, we need to identify two points on the graph. One way to do that is to choose some x-values, plug them into the equation, and find the corresponding y-values.
For example, when x = 0, y = 0 + 3 = 3. So we have the point (0, 3) on the graph.
Similarly, when x = -3, y = -3 + 3 = 0. So we have the point (-3, 0) on the graph.
Now, let's graph the second equation, -4x + 4y = 28. To find two points on this line, we can follow the same process of choosing x-values and calculating the corresponding y-values.
For x = 0, we have -4(0) + 4y = 28, which simplifies to 4y = 28, and y = 7. So we have the point (0, 7) on the graph.
For x = -7, we have -4(-7) + 4y = 28, which simplifies to 28 + 4y = 28, and 4y = 0, giving y = 0. So we have the point (-7, 0) on the graph.
Now, we can plot these points on the graph and draw a straight line passing through them.
After plotting the graph, we can observe that the two lines intersect at the point (-4, -1). This point represents the solution to the system of linear equations.
In summary, when we graph the given system of linear equations, we find that the lines intersect at the point (-4, -1). This tells us that (-4, -1) is the solution to the system of equations.
To graph the first equation y = x + 3, we can start by plotting two points and then connecting them. Let's choose x = 0. Substituting x = 0 into y = x + 3, we get y = 0 + 3 = 3. So, one point on the line is (0, 3). Let's choose another value for x, say x = 2. Substituting x = 2 into y = x + 3, we get y = 2 + 3 = 5. So, another point on the line is (2, 5). We can now plot these two points and connect them with a straight line:
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6 +
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5 + . (2, 5)
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4 +
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3 + . (0, 3)
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0 1 2 3 4 5 6
Now, we need to graph the second equation -4x + 4y = 28. To do this, we can solve for y:
-4x + 4y = 28
4y = 4x + 28
y = x + 7
We can use the same technique as before to find two points to plot. Let's choose x = 0. Substituting x = 0 into y = x + 7, we get y = 0 + 7 = 7. So, one point on the line is (0, 7). Let's choose another value for x, say x = 2. Substituting x = 2 into y = x + 7, we get y = 2 + 7 = 9. So, another point on the line is (2, 9). We can now plot these two points and connect them with a straight line:
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10 +
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9 +
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8 +
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7 + . (0, 7)
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6 +
|
-------------------
0 1 2 3 4 5 6
Now, looking at the graph, we can see that the two lines intersect at the point (4, 7). This means that the solution to the system of linear equations is x = 4 and y = 7.
To solve this system of linear equations by graphing, we need to graph each equation on the same coordinate plane and find the point(s) where the graphs intersect.
First, let's graph the equation y=x+3. To do this, we can start by finding the y-intercept, which is the value of y when x=0. In this case, when x=0, y=3. So one point on the graph is (0, 3).
Next, we can find another point on the graph by using the slope. The slope of this equation is 1, which means for every increase of 1 in x, y will increase by 1 as well. So if we move 1 unit to the right from the first point (0, 3), we get (1, 4).
Using these two points, we can draw a straight line connecting them. It should look something like this:
(Graph for y=x+3)
Now, let's graph the second equation: -4x+4y=28. To graph this equation, we can first rewrite it in slope-intercept form (y = mx + b), which looks like y = x + 7. To find points on this graph, we can start with the y-intercept, which is 7 in this case. So one point on the graph is (0, 7).
Next, using the slope, which is also 1, we can find another point. If we move 1 unit to the right from the first point (0, 7), we get (1, 8).
Using these two points, we can draw a straight line connecting them. It should look something like this:
(Graph for -4x+4y=28)
Now, we can observe the positions of the two lines on the graph. If the lines intersect at a single point, that will be the solution to the system of linear equations. However, if the lines are parallel and never intersect, or if they overlap completely, it means there are infinitely many solutions or no solution.
After graphing the two equations, we can see that the two lines intersect at a single point. This tells us that there is one unique solution to the system of linear equations.
To find the exact coordinates of the point of intersection, we can either measure it or solve the system of equations algebraically.