To estimate the solution to the system of the linear equations \( y = 5x + 10 \) and \( y = -2x + 3 \), we need to determine the point where these two lines intersect. Here's how you can do it:
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Graph the Equations:
- For the equation \( y = 5x + 10 \):
- This line has a slope of 5 and a y-intercept of 10, meaning it crosses the y-axis at (0, 10).
- For the equation \( y = -2x + 3 \):
- This line has a slope of -2 and a y-intercept of 3, meaning it crosses the y-axis at (0, 3).
- For the equation \( y = 5x + 10 \):
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Find Points:
- For the first equation, you can create a few points:
- When \( x = 0 \), \( y = 10 \) → Point (0, 10)
- When \( x = 1 \), \( y = 15 \) → Point (1, 15) (since \( 5(1) + 10 = 15 \))
- For the second equation, you can create points similarly:
- When \( x = 0 \), \( y = 3 \) → Point (0, 3)
- When \( x = 1 \), \( y = 1 \) → Point (1, 1) (since \( -2(1) + 3 = 1 \))
- For the first equation, you can create a few points:
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Estimate Intersection:
- Plot these points on a graph and draw the lines for both equations.
- The solution to the system of equations, or the intersection point, will be the point where these two lines cross.
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Solve Algebraically (if needed):
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To find where these lines intersect algebraically: \[ 5x + 10 = -2x + 3 \] \[ 5x + 2x = 3 - 10 \] \[ 7x = -7 \] \[ x = -1 \]
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Substitute \( x = -1 \) back into either equation to find \( y \): \[ y = 5(-1) + 10 = 5 \] or \[ y = -2(-1) + 3 = 5 \]
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Thus, the solution to the system of equations is \( (-1, 5) \). This point is where the two lines intersect on the graph.
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