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Solving Simultaneous Equations by Substitution (1)

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Simultaneous equations involve two variables, each of whose value must be found.

The following method, in which one equation is substituted into the other, works best when one equation contains a single letter on its own.

**Example**

Solve simultaneously

3x + 2y = 7

y = 4x - 2

**Answer**

3x + 2y = 7 (a)

y = 4x - 2 (b)

We substitute (b) into (a).

Wherever we see a y in (a) we will replace it with (4x - 2), remembering the brackets.

(a) becomes:

3x + 2(4x - 2) = 7

Multiply out the brackets and solve.

*Notice that we now have an equation just in x.*

3x + 8x - 4 = 7

11x - 4 = 7

11x = 11

x = 1

Now use (b) to determine y.

y = 4x - 2

When x = 1, this gives:

y = 4×1 - 2 = 4 - 2 = 2

y = 2

Check by putting this into the other equation (a) to get 3 × 1 + 2 × 2 = 3 + 4 = 7

Solution is **x = 1, y = 2**

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