Simultaneous equations

Simultaneous means happening at the same time.

An example of using the word, could be from an exciting spy novel. “To diffuse the bomb, the spy had to press the button and turn the key simultaneously.”

In simultaneous equations, there are two or more equations with two or more unknown values. The number of unknown values matches the number of equations.

If there are two unknown values in a single equation then it isn’t possible to work out the value of the unknowns.

For example, $y = 2x + 1$

There isn’t a single value for $x$ and a single value for $y$ that we can find by rearranging this equation. There are many solutions for $x$ and $y$ . For example, if $x= 1$ then $y= 3$ , if $x= 2$ then $y= 5$ .

If, at the same time, we had another equation that showed the relationship between $x$ and $y$ then we could find the value of $x$ and $y$ .

If as well as $y = 2x + 1$ , we knew that $y = 3x - 1$ then we could work out the unique values of $x$ and $y$ .

We can do this by substituting the value of $y$ given in one of the equations with the value of $y$ in the other.

The word substitute here is used in the same way as in a football match, where one player on the pitch is changed, or substituted, for another. Substitute means to put a person or thing in the place of another.

We start with $y = 2x + 1$ and $y = 3x - 1$

Then we substitute $2x + 1$ for $y$ in $y = 3x - 1$ as follows

$y = 3x - 1$

$2x + 1 = 3x - 1$

Now we have one equation with one unknown and we can solve it by rearranging.

$2x + 1 = 3x - 1$

Gather the $x$ terms together

$1 = x - 1$

Gather the numerical values

$2 = x$

$x=2$

We have worked out the value of $x$

We can then substitute this value for $x$ in either of the two original equations to get the value of $y$

$y = 2x + 1$

$y = 2 \times 2 + 1$

$y = 4 + 1$

$y = 5$

From the simultaneous equations, we have worked out that $x = 2$ and $y = 5$

We can check that we have this right by substituting for $x$ and $y$ in the second equation (the one we decided not to use to find out the value of $y$ )

$y = 3x - 1$

$5 = 3 \times 2 - 1$

$5 = 6 - 1$

$5 = 5$

This shows that we have worked out the values of $x$ and $y$ correctly.

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