Question

Solve each of the following pairs of simultaneous equations
1.x-5y=12
Y²=2x

Answer 1

To solve the pairs of simultaneous equations given by:

1. \( x - 5y = 12 \)
2. \( y^2 = 2x \)

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 \( y \):
1. \( y = \frac{24}{2} = 12 \)
2. \( y = \frac{-4}{2} = -2 \)

### Step 5: Solve for \( x \) using the values of \( y \)

Now, let's find the corresponding values of \( x \):

1. For \( y = 12 \):
\[
x = 5(12) + 12 = 60 + 12 = 72
\]
So, one solution is \( (72, 12) \).

2. 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:
1. \( (72, 12) \)
2. \( (2, -2) \)