Algebra

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Simultaneous Equations: Elimination 1

Solving simultaneous equations allows us to find unknown values which are common between two equations. These values are where two lines intersect on a graph.

Simultaneous equations are an algebraic method to work out where two lines intersect, and find solutions which are common to two equations.

Simultaneous equations enable us to find points which:

For example, let's try solving $3x + 4y = 22$ and $5x - 4y = -6$

Label these equations $1$ and $2$

$(1)\space3x + 4y = 22$ and $(2)\space5x - 4y = -6$

Notice a $+4y$ term and a $-4y$ term

$(1)\space3x \space{\color{#21affb}+ 4y} = 22$ and $(2)\space5x \space {\color{#21affb}- 4y} = -6$

We can eliminate $y$ by adding these together

By eliminating the $y$ terms, we can solve the equations to find $x$ .To add them together, we can combine them into a single equation and collect like terms together.

Add together $3x + 4y = 22$ and $5x - 4y = -6$

Solve this new equation $8x=16$

$x=2$ . We can now use this to find $y$

We can subsitute our result $x=2$ into one of the original equations to find the value of $y$ .

Find $y$ by putting $x=2$ into $3x + 4y = 22$

You've solved it!

$x = 2$ and $y = 4$

Both equations go through the point $(2,4)$

Well done!

Over to you! The next three questions will be a single example, but split into three parts. Remember the steps you've just taken! The full question is:

Solve $3x + 7y = 27$ and $-3x + 2y = 0$

Add $3x + 7y = 27$ and $-3x + 2y = 0$ together

$9y=27$ , so what is $y$ ?

$y= 3$ , so what is $x$ ? Equation (1) is $3x + 7y = 27$