Questions: Introduction to simultaneous equations
Before attempting these questions, it is highly recommended that you read Guide: Introduction to simultaneous equations.
Q1
Find how many solutions exist for the following sets of simultaneous equations.
1.1. \[\quad\begin{array}{ccc} x+2y & = & 4\\ 4x + 8y & = & 16 \end{array}\]
1.2. \[\quad\begin{array}{ccc} -2x+3y & = & 6\\ 4x - 6y & = & -12\end{array}\]
1.3. \[\quad\begin{array}{ccc} 3x+4y & = & 2\\ 8x + 2y & = & -1\end{array}\]
Q2
Using the substitution method, solve for \(x\) and \(y\) in the following pairs of simultaneous equations.
2.1. \[\quad\begin{array}{ccc} x+2y & = & -2\\ -4x - 6y & = & 4 \end{array}\]
2.2. \[\quad\begin{array}{ccc} 5x+y & = & 3\\ -10x - y & = & 7 \end{array}\]
2.3. \[\quad\begin{array}{ccc} -5x+y & = & 3\\ 3x + 2y & = & 12 \end{array}\]
2.4. \[\quad\begin{array}{ccc} 4x+3y & = & 20\\ 6x - 3y & = & 12 \end{array}\]
2.5. \[\quad\begin{array}{ccc} 7x-2y & = & 13\\ 2x + 3y & = & 17 \end{array}\]
2.6. \[\quad\begin{array}{ccc} 4x+y & = & 9\\ 9x - y & = & 4 \end{array}\]
2.7. \[\quad\begin{array}{ccc} 3y & = & 7-x\\ 3x & = & 4+y \end{array}\]
Q3
Using the elimination methods, solve for \(x\) and \(y\) in the following pairs of simultaneous equations.
3.1. \[\quad\begin{array}{ccc} x+3y & = & 7\\ 7x - 3y & = & 1 \end{array}\]
3.2. \[\quad\begin{array}{ccc} -x+4y & = & -13\\ 2x - 7y & = & 22 \end{array}\]
3.3. \[\quad\begin{array}{ccc} 8x+4y & = & 10\\ 2x - 5y & = & 3 \end{array}\]
3.4. \[\quad\begin{array}{ccc} 5x+6y & = & 19\\ 4x -9y & = & 6 \end{array}\]
3.5. \[\quad\begin{array}{ccc} 7x-3y & = & 20\\ 3x + 5y & = & 9 \end{array}\]
3.6. \[\quad\begin{array}{ccc} \dfrac{x}{2}+4y & = & 3\\[0.5em] \dfrac{y}{3} - 2x & = & 1 \end{array}\]
3.7. \[\quad\begin{array}{ccc} -y+1 & = & \dfrac{3x}{2}\\ 2x - \dfrac{y}{3} & = & 5 \end{array}\]
Q4
For the following sets of simultaneous equations, decide on the best method to use (between the substitution and elimination method) and solve for \(x\) and \(y\).
4.1. \[\quad\begin{array}{ccc} 5x + 2y & = & 7\\ 2x - y & = & 4 \end{array}\]
4.2. \[\quad\begin{array}{ccc} 3x + 4y & = & 12\\ 2x - 2y & = & 8 \end{array}\]
4.3. \[\quad\begin{array}{ccc} x - 7y & = & 5\\ 2x + 5y & = & 9 \end{array}\]
4.4. \[\quad\begin{array}{ccc} 4x + 3y & = & 10\\ 2x - 5y & = & -1 \end{array}\]
4.5. \[\quad\begin{array}{ccc} x-3y & = & 5\\ 2x + 5y & = & 9 \end{array}\]
After attempting the questions above, please click this link to find the answers.
Version history
v1.0: initial version created 12/24 by Ollie Brooke as part of a University of St Andrews VIP project.