To solve the system of equations graphically, we need to plot each equation on a coordinate plane.
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Equation 1: \(x + y = 0\)
- This can be rewritten as \(y = -x\).
- This is a straight line that passes through the origin (0, 0) and has a slope of -1.
- Two points on this line are:
- When \(x = 0\), \(y = 0\) (point: (0, 0)).
- When \(x = 4\), \(y = -4\) (point: (4, -4)).
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Equation 2: \(-3x + 4y = 14\)
- This can be rewritten as \(4y = 3x + 14\) or \(y = \frac{3}{4}x + \frac{14}{4}\) or \(y = \frac{3}{4}x + 3.5\).
- This is a straight line with a slope of \(\frac{3}{4}\) and a y-intercept of 3.5.
- Two points on this line can be found:
- When \(x = 0\), \(y = 3.5\) (point: (0, 3.5)).
- When \(x = 4\), \(y = \frac{3}{4}(4) + 3.5 = 3 + 3.5 = 6.5\) (point: (4, 6.5)).
Now, we can plot these points on the coordinate plane:
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For the first equation \(y = -x\):
- Plot (0, 0) and (4, -4), and draw the line through these points.
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For the second equation \(y = \frac{3}{4}x + 3.5\):
- Plot (0, 3.5) and (4, 6.5), and draw the line through these points.
Now, where the two lines intersect is the solution to the system of equations.
Solution:
When you plot these two lines on a graph, they will intersect at a single point. The coordinates of that intersection point will give you the solution to the system of equations.
To find the exact intersection point algebraically, you can solve the system.
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From \(x + y = 0\), we get \(y = -x\).
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Substitute \(y\) into the second equation:
\[ -3x + 4(-x) = 14 \] \[ -3x - 4x = 14 \] \[ -7x = 14 \implies x = -2 \]
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Substitute \(x = -2\) back into \(y = -x\):
\[ y = -(-2) = 2 \]
Thus, the solution to the system of equations is \( (x, y) = (-2, 2) \).
Summary:
- The equations are \(x + y = 0\) and \(-3x + 4y = 14\).
- The intersection point, found graphically and verified algebraically, is \((-2, 2)\).