Solve the simultaneous equations
x²+y²=26
x-y=4
From the second equation, we can solve for x:
x = y+4
Substituting this value of x into the first equation, we have:
(y+4)² + y² = 26
Expanding:
y² + 8y + 16 + y² = 26
Combining like terms:
2y² + 8y + 16 = 26
Subtracting 26 from both sides:
2y² + 8y - 10 = 0
Dividing both sides by 2:
y² + 4y - 5 = 0
Factoring:
(y + 5)(y - 1) = 0
Setting each factor equal to zero:
y + 5 = 0 or y - 1 = 0
Therefore, y = -5 or y = 1
If y = -5:
Substituting this value into the second equation to solve for x:
x - (-5) = 4
x + 5 = 4
Subtracting 5 from both sides:
x = -1
The solution for y = -5 is x = -1
If y = 1:
Substituting this value into the second equation to solve for x:
x - 1 = 4
Adding 1 to both sides:
x = 5
The solution for y = 1 is x = 5
Therefore, the simultaneous equations are solved when x = -1, y = -5 and when x = 5, y = 1.
To solve the simultaneous equations x² + y² = 26 and x - y = 4, we can use the method of substitution:
1. Rearrange the second equation to solve for x: x = y + 4.
2. Substitute this expression for x in the first equation: (y + 4)² + y² = 26.
3. Expand the equation: y² + 8y + 16 + y² = 26.
4. Combine like terms: 2y² + 8y + 16 = 26.
5. Subtract 26 from both sides: 2y² + 8y - 10 = 0.
6. Divide every term by 2 to simplify: y² + 4y - 5 = 0.
7. Factor the quadratic equation: (y - 1)(y + 5) = 0.
The solutions are y = 1 and y = -5.
8. Substitute these values back into the equation x = y + 4 to find the corresponding x-values:
- If y = 1, then x = 1 + 4 = 5.
- If y = -5, then x = -5 + 4 = -1.
So, the solution to the simultaneous equations is x = 5, y = 1 and x = -1, y = -5.