Question

How do you solve the system $4x - y = 3y + 7$ and $x + 8y = 4$ ?

Answer 1

The given equations are algebraic expressions as they contain both numerical values and alphabets in the equation. The alphabets represent some unknown quantities that can be found with the help of the algebraic expression. The given equations contain two unknown variable quantities x and y. We can find the value of these unknown quantities by methods like graphing, substitution method and elimination method.

Complete step by step answer:
We are given $4x - y = 3y + 7$ and $x + 8y = 4$
$4x - y = 3y + 7$ can be written as
$4x - y - 3y = 7 \\\ \Rightarrow 4x - 4y = 7\,\,...(1) \\\$
Second equation is $x + 8y = 4$
Multiplying both sides of this equation be 4, we get –
$4x + 32y = 16\,\,...(2)$
Now applying the elimination method, we subtract (2) from (1), as follows –
$4x + 32y = 16 \\\ \underline { - 4x + 4y = - 7} \\\ \underline {\,\,\,\,\,\,\,\,\,\,\,\,36y = 9\,\,} \\\$
$\Rightarrow y = \dfrac{9}{{36}} \\\ \Rightarrow y = \dfrac{1}{4} \\\$
Putting the value of y in (1), we get –
$4x - 4(\dfrac{1}{4}) = 7 \\\ \Rightarrow 4x - 1 = 7 \\\ \Rightarrow 4x = 8 \\\ \Rightarrow x = 2 \\\$
Hence for the system $4x - y = 3y + 7$ and $x + 8y = 4$ , we have $x = 2$ and $x = \dfrac{1}{4}$.

Note: For finding n number of unknown quantities we must have n equations, in this question we have to find two unknown quantities and we have exactly two equations to find their values so their values are obtained easily. If the number of equations is more than n, then the extra equation will be used for verifying the answers obtained. For applying the elimination method, we make the coefficient of one of the quantities equal by multiplying the two equations by some number, then we add or subtract the two equations to cancel out the variable whose coefficients are now equal, this way we
get the equation in terms of only one variable that can be solved easily. Thus, we can solve similar questions by following the above steps.