To solve the pairs of simultaneous equations given by:
- \( x - 5y = 12 \)
- \( y^2 = 2x \)
We will first express \( x \) in terms of \( y \) from the first equation and then substitute it into the second equation.
Step 1: Solve for \( x \) in terms of \( y \)
From the first equation: \[ x = 5y + 12 \]
Step 2: Substitute \( x \) into the second equation
Now we'll substitute \( x \) in the second equation \( y^2 = 2x \): \[ y^2 = 2(5y + 12) \] \[ y^2 = 10y + 24 \]
Step 3: Rearrange the equation
Next, rearrange this equation to set it to zero: \[ y^2 - 10y - 24 = 0 \]
Step 4: Factor or use the quadratic formula
We will use the quadratic formula to solve for \( y \): \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where \( a = 1, b = -10, c = -24 \). Calculating the discriminant: \[ b^2 - 4ac = (-10)^2 - 4(1)(-24) = 100 + 96 = 196 \] Now applying the quadratic formula: \[ y = \frac{10 \pm \sqrt{196}}{2} \] \[ y = \frac{10 \pm 14}{2} \] This gives us two possible solutions for \( y \):
- \( y = \frac{24}{2} = 12 \)
- \( y = \frac{-4}{2} = -2 \)
Step 5: Solve for \( x \) using the values of \( y \)
Now, let's find the corresponding values of \( x \):
-
For \( y = 12 \): \[ x = 5(12) + 12 = 60 + 12 = 72 \] So, one solution is \( (72, 12) \).
-
For \( y = -2 \): \[ x = 5(-2) + 12 = -10 + 12 = 2 \] So, the other solution is \( (2, -2) \).
Step 6: Summary of solutions
The pairs of solutions to the simultaneous equations are:
- \( (72, 12) \)
- \( (2, -2) \)