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In this video we are going to discover how to solve simultaneous equations by substitution. Simultaneous equations are two or more equations with two or more unknowns that must be solved at the same time; hence the name simultaneous.

There are three different methods for solving simultaneous equations: elimination, substitution and graphically. Elimination doesn’t always work and solving graphically is slow and not accurate. Solving them using substitution does always work, which we will look at in this video.

When we solve simultaneous equations, we may get one solution, like with linear equations, or if a quadratic is involved, we may get two solutions. And we might also get 2 solutions with a straight line and a circle. Before we start, we need to know what linear equations look like: they are equations with an ‘x’ and a ‘y’ in, or an ‘a’ and ‘b’, but no squared or cubed letters.

To solve simultaneous equations by substitution, start by rearranging the linear equation to become y=. E.g. y = x + 3. We can now substitute this y= into the non-linear equation. So in place of the y in the non-linear equation, we write x+3 instead. x + 3 = x^2 + 1. Rearrange this to get a quadratic, then factorise and solve it. This gives two x values: x = -1 and x = 2. Because we have 2 x-values, we’ll need 2 y-values. Substitute the x-values into the linear y equals equation y = x + 3. So when x = -1, y = 2. And when x = 2, y = 5. So the solutions are (-1, 2) and (2, 5).

As always with maths, we should double check our answers. Substitute the solutions (-1, 2) and (2, 5) into the non-linear equation that was given in the question.

So there we have solving simultaneous equations by substitution. Take your time with the algebra, and always double check your answer at the end. Substitution will always work, whereas elimination won’t always.

Which is why some people choose to use substitution for any simultaneous equations, even if they are easy linear ones.

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