Proportionality allowed us to investigate 2 variables linked to each other in some way, simultaneous equations are 2 equations that are linked to each other in some way that we need to investigate.
There are 2 main strategies to solve simultaneous equations. The first method is to combine them into 1 equation which is when we substitute values from 1 equation into the other. This involves us rearranging the equation to usually make a letter the subject of the equation then putting the other side into the letter of the other equation. This is the hardest method and we would recommend you to try the other method on questions before trying this one. Let’s try an example:
$$ x + y = 14 \text{ \color{blue}{ (1)}} $$ $$ x - y \color{red}{+y} = 2 \color{red}{+y} \text{ \color{blue}{ (2)}} $$ $$ x = 2 + y $$ $$ \text{ \color{blue}{ (1) into (2)}} $$ $$ \color{red}{2 + y} + y = 14 = 2y + 2 $$ $$ 2y + 2 \color{red}{-2} = 14 \color{red}{-2} $$ $$ \dfrac{12}{\color{red}{2}} = \dfrac{2y}{\color{red}{2}} \therefore y = 6 $$
Now that we have a value for y we can substitute that into the other equation. To point out, it’s probably a good idea to label each equation so you don’t confuse which is which:
$$ \text{ \color{blue}{ (2) into (1)}} $$ $$ x - \color{red}{6} \color{blue}{+ 6} = 2 \color{blue}{+ 6} $$ $$ x = 8 $$
Remember, as always to check your result by plugging the answers into the initial equation!
An easier way to do this would be the elimination method which is when we subtract or add one entire equation from the other to eliminate one unknown letter to leave us with only 1. Often students will find one value via the elimination method and then find the other by substitution. However, in this example we will show using the elimination method to find both values.
Firstly, we must make each equation contain the same amount of a certain letter then add or subtract away that letter leaving the rest:
$$ 2x + 3y = 17 \text{\color{blue}{ (1) }} \color{red}{\times 3} $$ $$ 3x - 5y = 35 \text{\color{blue}{ (2) }} \color{red}{\times 2} $$ $$ 6x - 10y = 70 \text{\color{blue}{ (2) }} $$ $$ 6x + 9y = 51 \text{\color{blue}{ (1) }} $$ $$ \text{\color{blue}{ (2) - (1) }} $$ $$ -19y = 19 \therefore y = -1 $$
You could then find the value for x by substituting the value we have for y into an equation. However, we will use elimination again:
$$ 2x + 3y = 17 \text{\color{blue}{ (1) }} \color{red}{\times 5} $$ $$ 3x - 5y = 35 \text{\color{blue}{ (2) }} \color{red}{\times 3} $$ $$ 10x + 15y = 85 \text{\color{blue}{ (1) }} $$ $$ 9x - 15y = 105 \text{\color{blue}{ (2) }} $$ $$ \text{\color{blue}{ (1) + (2) }} $$ $$ 19x = 190 \therefore x = 10 $$
Probably the coolest thing about simultaneous equations is the last method to solve them. Which is that the solution to y and x is the point where the lines cross each other on a graph (intersect).
As we can see both of the lines meet at the coordinates:
$$ (x, y) \Rightarrow (10, -1) $$
So the point of intersection of the 2 lines is the solution for x and y in both equations.
More coming soon