Question
Estimate the solution to the system of these linear equations based on the graph.
y = -x + 6
y = 3x - 2
y = -x + 6
y = 3x - 2
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GPT-4 Turbo
To estimate the solution to the system of linear equations, we first need to understand what the equations represent graphically.
1. y = -x + 6 represents a straight line with a slope of -1 and a y-intercept of 6. This means the line will slope downwards to the right and will cross the y-axis at the point (0, 6).
2. y = 3x - 2 represents another straight line with a slope of 3 and a y-intercept of -2. This line will slope upwards to the right and will cross the y-axis at the point (0, -2).
The solution to the system of equations is the point where both lines intersect. Graphically, we can estimate this point by plotting both lines on the same set of axes and looking for their point of intersection.
Without an actual graph to look at, I can only provide you with a way to plot the lines roughly on your own. If you draw the y-axis and x-axis on a piece of paper, you can plot the y-intercepts of both lines:
- For y = -x + 6, plot the point (0, 6) on the y-axis.
- For y = 3x - 2, plot the point (0, -2) on the y-axis.
Next, you can find another point on each line by choosing a value for x and solving for y. For example:
- For y = -x + 6, if you choose x = 6, then y = -6 + 6 = 0. So another point on this line is (6, 0).
- For y = 3x - 2, if you choose x = 1, then y = 3(1) - 2 = 1. So another point on this line is (1, 1).
Draw each line through its respective points, and the lines will intersect at the solution to the system.
Since both lines are linear and have different slopes, they will intersect at exactly one point, which represents a unique solution to the system.
To find the exact solution algebraically, you need to set the equations equal to one another and solve for x:
- -x + 6 = 3x - 2
Add x to both sides:
- 6 = 4x - 2
Add 2 to both sides:
- 8 = 4x
Divide by 4:
- 2 = x
Now that we have the x-coordinate of the solution, we can substitute it back into either of the original equations to find the y-coordinate:
Using y = -x + 6:
- y = -2 + 6
- y = 4
So the exact solution is (2, 4), and graphically, the two lines should intersect at this point.
1. y = -x + 6 represents a straight line with a slope of -1 and a y-intercept of 6. This means the line will slope downwards to the right and will cross the y-axis at the point (0, 6).
2. y = 3x - 2 represents another straight line with a slope of 3 and a y-intercept of -2. This line will slope upwards to the right and will cross the y-axis at the point (0, -2).
The solution to the system of equations is the point where both lines intersect. Graphically, we can estimate this point by plotting both lines on the same set of axes and looking for their point of intersection.
Without an actual graph to look at, I can only provide you with a way to plot the lines roughly on your own. If you draw the y-axis and x-axis on a piece of paper, you can plot the y-intercepts of both lines:
- For y = -x + 6, plot the point (0, 6) on the y-axis.
- For y = 3x - 2, plot the point (0, -2) on the y-axis.
Next, you can find another point on each line by choosing a value for x and solving for y. For example:
- For y = -x + 6, if you choose x = 6, then y = -6 + 6 = 0. So another point on this line is (6, 0).
- For y = 3x - 2, if you choose x = 1, then y = 3(1) - 2 = 1. So another point on this line is (1, 1).
Draw each line through its respective points, and the lines will intersect at the solution to the system.
Since both lines are linear and have different slopes, they will intersect at exactly one point, which represents a unique solution to the system.
To find the exact solution algebraically, you need to set the equations equal to one another and solve for x:
- -x + 6 = 3x - 2
Add x to both sides:
- 6 = 4x - 2
Add 2 to both sides:
- 8 = 4x
Divide by 4:
- 2 = x
Now that we have the x-coordinate of the solution, we can substitute it back into either of the original equations to find the y-coordinate:
Using y = -x + 6:
- y = -2 + 6
- y = 4
So the exact solution is (2, 4), and graphically, the two lines should intersect at this point.
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