Syllabus

solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic

e.g. \begin{cases} x + y + 1 = 0 \\ x^2 + y^2 = 25 \end{cases}, \begin{cases} 2x + 3y = 7 \\ 3x^2 = 4 + 4xy \end{cases}

To solve a pair of simultaneous equations of which one is linear and one is quadratic using substitution, we follow these steps:

Example: \begin{cases}x + y = 7 \\ x^2 + y^2 = 37 \end{cases}

1. Solve one of the equations for one of the variables in terms of the other variable. It is usually easier to solve the linear equation for one of the variables.
   We can solve for y in terms of x by subtracting x from both sides: y = 7 - x.

2. Substitute the expression obtained in step 1 into the other equation, replacing the variable that has been eliminated.
   x^2 + (7 - x)^2 = 37

3. Solve the resulting quadratic equation for the remaining variable.

\begin{aligned}
x^2 - 14x + 49 + x^2 & = 37 \\
2x^2 - 14x + 12 & = 0 \\
 x = 1 \text{ or } x = 6
\end{aligned}

4. Substitute the value obtained in step 3 into the equation from step 1 to obtain the value of the other variable.
   If x = 1, then y = 7 - 1 = 6.
   If x = 6, then y = 7 - 6 = 1.

5. Check the solution by substituting both values into both original equations to verify that they are true.