Question

How do you solve the system of equations $8x - 2y = 48$and $3x - 4y = 1?$

Answer 1

Here we are given two sets of equations, first of all we will convert the coefficient of any of the variables common in both the set of equations. Then will use the elimination method to get the values of x and y.

Complete step-by-step solution:
Take the given expressions:
$8x - 2y = 48$
Multiply the above expression by
$16x - 4y = 96$ …. (A)
$3x - 4y = 1$ ….. (B)
Subtract equation (B) from (A)
Subtract the left hand side of both the equations and the right hand side of the equations.
$(16x - 4y) - (3x - 4y) = 96 - 1$
When there is a negative sign outside the bracket then the sign of the terms inside the bracket changes. Positive terms become negative and vice-versa.
$16x - 4y - 3x + 4y = 95$
Like values with the same value and opposite sign cancels each other.
$\Rightarrow 16x - 3x = 95$
Simplify the above expression –
$\Rightarrow 13x = 95$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$\Rightarrow x = \dfrac{{95}}{{13}}$
Place the above value in equation (B)
$\Rightarrow 3\left( {\dfrac{{95}}{{13}}} \right) - 4y = 1$
Simplify the above equation –
$\Rightarrow \left( {\dfrac{{285}}{{13}}} \right) - 1 = 4y$
Simplify finding the LCM (least common multiple)
$\Rightarrow \left( {\dfrac{{285 - 13}}{{13}}} \right) = 4y \\\ \Rightarrow \left( {\dfrac{{272}}{{13}}} \right) = 4y \\\ \Rightarrow y = \dfrac{{272}}{{13 \times 4}} \\\ \Rightarrow y = \dfrac{{68}}{{13}} \\\$

Hence, the required solution – $(x,y) = \left( {\dfrac{{95}}{{13}},\dfrac{{68}}{{13}}} \right)$.

Note:
Always remember that when we expand the brackets or open the brackets, sign outside the bracket is most important. If there is a positive sign outside the bracket then the values inside the bracket does not change and if there is a negative sign outside the bracket then all the terms inside the bracket changes. Positive terms change to negative and negative term changes to positive. While doing simplification remember the golden rules-
- Addition of two positive terms gives the positive term
- Addition of one negative and positive term, you have to do subtraction and give sign of bigger numbers whether positive or negative.
- Addition of two negative numbers gives a negative number but in actual you have to add both the numbers and give a negative sign to the resultant answer.