There are two ways to handle simultaneous equations.
This is the addition (or subtraction) method. What this involves is adding (or subtracting) the simultaneous equations together. The other way is the substitution method.
Manipulate one of the equations so that a pair of similar terms in each equation (either both `x`s or both `y`s)can be added or subtracted together to make them disappear. You will then be left with an equation that has a single term, and that can be solved as a linear equation.
Once the linear equation has been solved, substitute back into one of the original equations to get the value of the other variable.
If multiplying by one of the equations up to make the `x` or `y` values equal, then make sure that you multiply ALL the terms in the equation being multiplied.
Finally, check with the other equation that the solutions are correct.
What are the values of `x` and `y` in the simultaneous equations `2x+3y=14` and `6x-3y=18`?
| Equation 1 | `2x` | `+` | `3y` | `=` | `14` |
| Equation 2 | `6x` | `+` | `-3y` | `=` | `18` |
| Add the terms together | `2x+6x` | `+` | `3y-3y` | `=` | `14+18` |
| Which is | `8x` | `=` | `32` | ||
| Dividing both sides by 8 | `x` | `=` | `4` | ||
| We now have a value for `x` | |||||
| Take 1st equation | `2x` | `+` | `3y` | `=` | `14` |
| Substitute for `x` | `2(4)` | `+` | `3y` | `=` | `14` |
| `8` | `+` | `3y` | `=` | `14` | |
| Subtract 8 from both sides | `3y` | `=` | `6` | ||
| Divide both sides by 3 | `y` | `=` | `2` | ||
| We now have a value for `y` | |||||
| Check with equation 2 | 6(4) | `-` | `3(2)` | `=` | `18` ✔ |
Answer: `x=4` and `y=2`
Find the value for `x` that satisfies the simultaneous equations `4x+3y=14` and `7x+6y=23`
| Equation 1: | `4x` | `+` | `3y` | `=` | `14` |
| Equation 2: | `7x` | `+` | `6y` | `=` | `23` |
Multiply 1st equation by a value of 2 so that the `y` coefficients are equal.
Note that EACH term in the equation must be multiplied by 2
| Multiplied by 2 | `8x` | `+` | `6y` | `=` | `28` |
| `7x` | `+` | `6y` | `=` | `23` | |
| Subtract equation 2 from equation 1: | `8x-7x` | `+` | `6y-6y` | `=` | `28-23` |
| Which is | `x` | `=` | `5` | ||
| Substituting into equation 1 | `4(5)` | `+` | `3y` | `=` | `14` |
| `20` | `+` | `3y` | `=` | `14` | |
| Subtract 20 from both sides | `3y` | `=` | `-6` | ||
| Divide both sides by 3 | `y` | `=` | `-2` | ||
| Check with equation 2 | 7(5) | `+` | `6(-2)` | `=` | `23` ✔ |
Answer: `x=5`
See also Adding and Subtracting Terms and Solving Linear Equations