To solve the equations given: 1. **Equations**: - \( m^2 + n^2 = 29 \) (Equation 1) - \( m + n = 7 \) (Equation 2) We need to find the values of \( m \) and \( n \) that satisfy both equations. ### Step 1: Express one variable in terms of the other From Equation 2, we can express \( n \) in terms of \( m \): \[ n = 7 - m \] ### Step 2: Substitute into the first equation Now, we substitute \( n \) into Equation 1: \[ m^2 + (7 - m)^2 = 29 \] ### Step 3: Expand the equation Expanding \( (7 - m)^2 \): \[ (7 - m)^2 = 49 - 14m + m^2 \] Now substitute this back into the equation: \[ m^2 + (49 - 14m + m^2) = 29 \] Combine like terms: \[ 2m^2 - 14m + 49 = 29 \] ### Step 4: Simplify the equation Subtract 29 from both sides: \[ 2m^2 - 14m + 20 = 0 \] ### Step 5: Divide the entire equation by 2 To simplify, divide everything by 2: \[ m^2 - 7m + 10 = 0 \] ### Step 6: Factor the quadratic equation Now we need to factor the quadratic: \[ (m - 5)(m - 2) = 0 \] Setting each factor to zero gives us: \[ m - 5 = 0 \quad \Rightarrow \quad m = 5 \] \[ m - 2 = 0 \quad \Rightarrow \quad m = 2 \] ### Step 7: Find corresponding values of \( n \) Now we can find the corresponding values of \( n \) using \( n = 7 - m \): - If \( m = 5 \): \[ n = 7 - 5 = 2 \quad \Rightarrow \quad (m, n) = (5, 2) \] - If \( m = 2 \): \[ n = 7 - 2 = 5 \quad \Rightarrow \quad (m, n) = (2, 5) \] ### Step 8: List the solutions Thus, the pairs \( (m, n) \) that satisfy both equations are: - \( (5, 2) \) - \( (2, 5) \) ### Step 9: Evaluate the options Now, let's evaluate the options provided: - **A. (2, 3) and (3, 5)**: Incorrect, as neither pair satisfies both equations. - **B. (2, 5) and (5, 2)**: Correct, as both pairs satisfy the equations. - **C. (5, 2) and (5, 3)**: Incorrect, as (5, 3) does not satisfy the first equation. - **D. (5, 3) and (3, 5)**: Incorrect, as neither pair satisfies both equations. ### Conclusion The correct option is **B. (2, 5) and (5, 2)**. ### Revision Summary - To solve simultaneous equations, express one variable in terms of the other and substitute. - Expand and simplify the equations to form a quadratic equation. - Factor the quadratic to find the values of the variables. - Verify the solutions against the original equations to ensure they are correct.