Simultaneous equations: solved examples

Worked examples of simultaneous equations, from easy to hard, with every step written out. Try each one yourself first, then check the working.

✓ Every answer here is computed and checked by the math engine

Easy

Example 1. Solve the simultaneous equations $x + y = -2$ and $x - y = 4$ .
1. Eliminate one variable and solve for the other. $\begin{cases} x + y = -2 \\ x - y = 4 \end{cases}$
2. Solve and check both equations. $x = 1,\quad y = -3$
Answer: $x = 1,\ y = -3$
Example 2. Solve the simultaneous equations $x + y = -9$ and $x - y = -3$ .
1. Eliminate one variable and solve for the other. $\begin{cases} x + y = -9 \\ x - y = -3 \end{cases}$
2. Solve and check both equations. $x = -6,\quad y = -3$
Answer: $x = -6,\ y = -3$
Example 3. Solve the simultaneous equations $x + y = 9$ and $x - y = -1$ .
1. Eliminate one variable and solve for the other. $\begin{cases} x + y = 9 \\ x - y = -1 \end{cases}$
2. Solve and check both equations. $x = 4,\quad y = 5$
Answer: $x = 4,\ y = 5$

Example 4. Solve the simultaneous equations $5x + 3y = -4$ and $4x + 2y = -2$ .
1. Eliminate one variable and solve for the other. $\begin{cases} 5x + 3y = -4 \\ 4x + 2y = -2 \end{cases}$
2. Solve and check both equations. $x = 1,\quad y = -3$
Answer: $x = 1,\ y = -3$
Example 5. Solve the simultaneous equations $x + 5y = -21$ and $2x + 5y = -27$ .
1. Eliminate one variable and solve for the other. $\begin{cases} x + 5y = -21 \\ 2x + 5y = -27 \end{cases}$
2. Solve and check both equations. $x = -6,\quad y = -3$
Answer: $x = -6,\ y = -3$
Example 6. Solve the simultaneous equations $x + 5y = 29$ and $3x + y = 17$ .
1. Eliminate one variable and solve for the other. $\begin{cases} x + 5y = 29 \\ 3x + y = 17 \end{cases}$
2. Solve and check both equations. $x = 4,\quad y = 5$
Answer: $x = 4,\ y = 5$

Example 7. Solve the simultaneous equations $8x + 3y = -1$ and $-4x - 5y = 11$ .
1. Eliminate one variable and solve for the other. $\begin{cases} 8x + 3y = -1 \\ -4x - 5y = 11 \end{cases}$
2. Solve and check both equations. $x = 1,\quad y = -3$
Answer: $x = 1,\ y = -3$
Example 8. Solve the simultaneous equations $-7x + 6y = 24$ and $-4x + 7y = 3$ .
1. Eliminate one variable and solve for the other. $\begin{cases} -7x + 6y = 24 \\ -4x + 7y = 3 \end{cases}$
2. Solve and check both equations. $x = -6,\quad y = -3$
Answer: $x = -6,\ y = -3$
Example 9. Solve the simultaneous equations $-9x + 9y = 9$ and $-x - 8y = -44$ .
1. Eliminate one variable and solve for the other. $\begin{cases} -9x + 9y = 9 \\ -x - 8y = -44 \end{cases}$
2. Solve and check both equations. $x = 4,\quad y = 5$
Answer: $x = 4,\ y = 5$

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