Let's solve the system of simultaneous equations step by step.
The given equations are:
1. [tex]\(y = x + 4\)[/tex]
2. [tex]\(x^2 + y^2 = 40\)[/tex]
Step 1: Substitute the expression for [tex]\(y\)[/tex] from equation 1 into equation 2.
[tex]\[ y = x + 4 \][/tex]
Substituting [tex]\(y\)[/tex] into [tex]\(x^2 + y^2 = 40\)[/tex]:
[tex]\[ x^2 + (x + 4)^2 = 40 \][/tex]
Step 2: Expand the squared term in the equation.
[tex]\[ x^2 + (x^2 + 8x + 16) = 40 \][/tex]
[tex]\[ x^2 + x^2 + 8x + 16 = 40 \][/tex]
[tex]\[ 2x^2 + 8x + 16 = 40 \][/tex]
Step 3: Simplify and solve the resulting quadratic equation.
[tex]\[ 2x^2 + 8x + 16 - 40 = 0 \][/tex]
[tex]\[ 2x^2 + 8x - 24 = 0 \][/tex]
[tex]\[ x^2 + 4x - 12 = 0 \][/tex]
Step 4: Factor the quadratic equation.
[tex]\[ (x + 6)(x - 2) = 0 \][/tex]
Step 5: Solve for [tex]\(x\)[/tex].
The solutions for [tex]\(x\)[/tex] are:
[tex]\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \][/tex]
[tex]\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \][/tex]
Step 6: Use the values of [tex]\(x\)[/tex] to find the corresponding values of [tex]\(y\)[/tex] from equation 1.
For [tex]\(x = -6\)[/tex]:
[tex]\[ y = -6 + 4 = -2 \][/tex]
For [tex]\(x = 2\)[/tex]:
[tex]\[ y = 2 + 4 = 6 \][/tex]
Step 7: Write the solutions as ordered pairs [tex]\((x, y)\)[/tex].
The solutions to the system of equations are:
[tex]\[ (-6, -2) \][/tex]
[tex]\[ (2, 6) \][/tex]
Therefore, the solutions to the given system of equations are:
[tex]\[ \boxed{(-6, -2), (2, 6)} \][/tex]
