To review the system of two linear equations and find the solution, we can write down the equations first:
- \( x - 5y = 7 \) (Equation 1)
- \( 3x - 2y = 4 \) (Equation 2)
Finding the Intersection Point
The solution (or intersection point) of these equations can be found using various methods such as substitution, elimination, or graphing. Here, I will use the substitution method.
Step 1: Solve the first equation for \( x \)
From Equation 1: \[ x = 5y + 7 \]
Step 2: Substitute \( x \) into the second equation
Now substitute \( x \) in Equation 2: \[ 3(5y + 7) - 2y = 4 \]
Expanding the equation: \[ 15y + 21 - 2y = 4 \] \[ 13y + 21 = 4 \]
Step 3: Solve for \( y \)
Now, isolate \( y \): \[ 13y = 4 - 21 \] \[ 13y = -17 \] \[ y = -\frac{17}{13} \]
Step 4: Substitute \( y \) back into the expression for \( x \)
Now plug \( y \) back into the expression for \( x \): \[ x = 5\left(-\frac{17}{13}\right) + 7 \] \[ x = -\frac{85}{13} + \frac{91}{13} \] \[ x = \frac{6}{13} \]
Intersection Point
The intersection point of the two equations, which is the solution to the system, is: \[ \left( \frac{6}{13}, -\frac{17}{13} \right) \]
Summary
- The solution to the system of equations \( x - 5y = 7 \) and \( 3x - 2y = 4 \) is \( \left( \frac{6}{13}, -\frac{17}{13} \right) \). This point is where the two lines represented by the equations intersect on a graph.
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