# How do you solve the following system:  3x – 5y = 53 ,5x - 7y = 12 ?

Mar 9, 2017

Why on earth do they chose such awful numbers????

$x = - 77.75$
$y = - 57.25$

#### Explanation:

In solving equations for variable values the method is to manipulate so that you have a load of values and just one variable. It is then solvable. Variables in this case being $x \mathmr{and} y$.

$3 x - 5 y = 53 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots . . E q u a t i o n \left(1\right)$
$5 x - 7 y = 12 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots . . E q u a t i o n \left(2\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Solving for } x}$

Consider equation(1)

To make the -5y positive multiply everything by -1

$- 3 x + 5 y = - 53$

Add $3 x$ to both sides

$\textcolor{g r e e n}{- 3 x \textcolor{red}{+ 3 x} + 5 y = - 53 \textcolor{red}{+ 3 x}}$

But $- 3 x + 3 x = 0$

$0 + 5 y = 3 x - 53$

Divide both sides by 5 ( this is the same as $\textcolor{red}{\times \frac{1}{5}} \text{ }$)

color(green)(5/(color(red)(5))y=3/(color(red)(5))x-53/(color(red)(5))

But $\frac{5}{5} = 1$

$y = \frac{3}{5} x - \frac{53}{5} \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots E q u a t i o n \left({1}_{a}\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Using $E q u a t i o n \left({1}_{a}\right)$ substitute for y in $E q u a t i o n \left(2\right)$

$5 x - 7 y = 12 \text{ "->" } 5 x - 7 \left(\frac{3}{5} x - \frac{53}{5}\right) = 12$

$\text{ "25/5x" } - \frac{21}{5} x + \frac{371}{5} = 12$

$\text{ } \frac{4}{5} x = 12 - \frac{371}{5}$

$\text{ } 4 x = 60 - 371$

$\text{ } \textcolor{b l u e}{x = - \frac{311}{4}}$

$\textcolor{b l u e}{\text{This is the same as } x = - 77.75}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Solving for } y}$

Substitute for $x$ in $E q u a t i o n \left(2\right)$
It does not matter which of the 2 equations you use.

$5 x - 7 y = 12 \text{ "->" } 5 \left(- \frac{311}{4}\right) - 7 y = 12$

$\text{ } - \frac{1555}{4} - 7 y = 12$

$\text{ } - \frac{1555}{4} - \frac{28}{4} y = \frac{48}{4}$

$\text{ } 28 y = - 1555 - 48$

$\text{ } y = - \frac{229}{4}$

$\textcolor{b l u e}{\text{This is the same as } y = - 57.25}$

Mar 9, 2017

$x = - 77.75 \mathmr{and} y = - 57.25$

#### Explanation:

There are several methods for solving a system of equations, also known as simultaneous equations. The method you choose will depend on the format of the equations you are given.

• Substitution - [write one variable in terms of the other]
• Equating - [one variable is expressed in two different ways]
• Matrices - [ multiply a matrix by its inverse to get the unit matrix]
• Graphically - [ graph each equation and find the intersection]

In this example I will use the elimination method.

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots} 3 x - 5 y = 53 \textcolor{w h i t e}{\ldots \ldots \ldots \ldots .} A$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots} 5 x - 7 y = 12 \textcolor{w h i t e}{\ldots \ldots \ldots \ldots .} B$

$A \times 5 : \text{ } \textcolor{red}{15 x} - 25 y = 265 \textcolor{w h i t e}{\ldots \ldots \ldots .} C$
$B \times \text{-3: } \textcolor{red}{- 15 x} + 21 y = - 36 \textcolor{w h i t e}{\ldots \ldots .} D$

$C + D : \textcolor{w h i t e}{\ldots \ldots \ldots \ldots .} - 4 y = 229 \text{ }$ there is no x-term!
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} \textcolor{b l u e}{y = - 57.25}$

Now that you have value for $\textcolor{b l u e}{y}$, substitute it into any of the equations above to find $x$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots} 5 x - 7 \textcolor{b l u e}{y} = 12 \textcolor{w h i t e}{\ldots \ldots \ldots \ldots .} B$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots .} 5 x - 7 \textcolor{b l u e}{\left(\text{-57.25}\right)} = 12$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots .} 5 x \textcolor{b l u e}{\text{+400.75}} = 12$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots . .} 5 x \text{ } = - 388.75$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots .} x \text{ } = - 77.75$
.

[Note that: $\textcolor{red}{15 x} \mathmr{and} \textcolor{red}{- 15 x}$ are additive inverses. They therefore ADD to give 0.]

This was done by multiplying B by $- 3$ to change the sign.