Simultaneous Equations
Simultaneous equations involve solving two or more equations together to find values that satisfy all of them at the same time. Students will learn two methods, substitution and elimination.
The elimination method removes one variable by adding or subtracting the equations.
The substitution method involves expressing one variable in terms of another and replacing it into the second equation.
The elimination method is effective for solving simultaneous equations with certain structures, while the substitution method can be applied to all types of simultaneous equations. Therefore, mastering the substitution method is the most important.
Related Lessons:
Elimination Setup
When using the elimination method, it is important to first arrange the two equations neatly so that like terms are aligned. This means placing the \(x\) terms in one column, the \(y\) terms in another column, and ensuring the “=” signs and constant values are directly in line with each other.
By setting up the equations in this structured way, it becomes much easier to add or subtract them accurately to eliminate one variable and reduce careless mistakes.
Elimination Method
Once we have properly set up the simultaneous equations, we can proceed to eliminate one variable by adding or subtracting the equations. This step allows us to simplify the system and solve it more easily. However, one limitation of the elimination method is that it is mainly used for solving linear equations and is not suitable for non-linear equations.
Substitution Method
The substitution method is widely used for solving all types of simultaneous equations, and it is an essential technique that every student should master. To use this method, we first rearrange one equation to make one variable the subject in terms of the other. Next, we substitute this expression into the second equation and simplify to find the value of one variable. Once we have this value, we can substitute it back into either equation to find the remaining unknown.
Simultaneous Equation Example 1
Solve the following equation:
\[
\begin{align*}
2𝑦−𝑥&=4\\
3𝑥+4𝑦&=−7\\
\end{align*}
\]
Simultaneous Equation Example 2
Solve the following equation:
\[
\begin{align*}
4𝑥−3𝑦&=8\\
6𝑥+𝑦&=1\\
\end{align*}
\]
Simultaneous Equation Example 3
Solve the following equation:
\[
\begin{align*}
4𝑥+𝑦&=1\\
2𝑥+3𝑦&=13\\
\end{align*}
\]