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Solve the system by graphing. \( y=-1 \) \( y=-\frac{5}{2} x+4 \) Solution: x-coordinate: \( y \)-coordinate:

Ask by Ramos Delgado. in the United States
Jan 21,2025

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Tutor-Verified Answer

Answer

The solution is the point \((2, -1)\).

Solution

To solve the system of equations by graphing, we need to find the point where the two lines intersect. **Given Equations:** 1. \( y = -1 \) This is a horizontal line that crosses the y-axis at \(-1\). 2. \( y = -\frac{5}{2}x + 4 \) This is a straight line with a slope of \(-\frac{5}{2}\) and a y-intercept at \(4\). **Finding the Intersection Point:** Set the two equations equal to each other to find the x-coordinate of the intersection: \[ -1 = -\frac{5}{2}x + 4 \] **Step-by-Step Solution:** 1. **Subtract 4 from both sides:** \[ -1 - 4 = -\frac{5}{2}x \] \[ -5 = -\frac{5}{2}x \] 2. **Multiply both sides by \(-\frac{2}{5}\) to solve for \(x\):** \[ x = \left(-5\right) \times \left(-\frac{2}{5}\right) = 2 \] 3. **Substitute \(x = 2\) back into one of the original equations to find \(y\):** \[ y = -1 \] **Solution:** - **x-coordinate:** \(2\) - **y-coordinate:** \(-1\) **Intersection Point:** \((2, -1)\)

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Bonus Knowledge

To solve the system by graphing, you would plot the line \( y = -1 \), which is a horizontal line crossing the y-axis at -1, indicating that for any value of \( x \), \( y \) will always be -1. Next, plot the line \( y = -\frac{5}{2} x + 4 \). This line has a y-intercept of 4 and a slope that indicates it falls steeply. The two lines intersect at a point, which represents the solution to the system. To find the coordinates of the intersection, set \( y = -1 \) equal to the second equation: \(-1 = -\frac{5}{2} x + 4\) Rearranging gives: \(-\frac{5}{2} x = -1 - 4\) \(-\frac{5}{2} x = -5\) Multiplying both sides by \(-\frac{2}{5}\) results in: \( x = 2 \) Now substituting \( x = 2 \) back into one of the original equations to find \( y \): \( y = -1 \) Thus, the solution is: x-coordinate: \( 2 \) y-coordinate: \( -1 \) And there you have it!

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