Question

How do you put \[7x + 14y = 3x - 10\;\] into standard form?

Answer 1

Hint: The standard form of a line is in the form \[Ax + By = C\] where\[A,B,and\;C\]are integers. The standard form of a line is just another way of writing the equation of a line. It gives all of the same information as the slope-intercept form. Using the above information we can solve the given question.

Complete step-by-step solution:
Given
\[7x + 14y = 3x - 10\;.........................\left( i \right)\]
We have given an equation which has to be represented in the standard form which is \[Ax + By = C\] where \[A,B,and\;C\] are integers.
\[Ax + By = C........................\left( {ii} \right)\]
So we have to convert the given expression in (i) to the general form using various mathematical operations.
Such that we have to manipulate the given equation to the standard form which can be achieved by performing different arithmetic operations on both LHS and RHS equally.
So first let’s subtract the term$3x$from both LHS and RHS.
We get:
\[
  \Rightarrow 7x + 14y = 3x - 10\; \\
 \Rightarrow 7x - 3x + 14y = 3x - 3x - 10\; \\
 \Rightarrow 4x + 14y = - 10........................\left( {iii} \right)\; \\
 \]
Now on observing (iii) we can say that it can be further simplified such that:
\[
\Rightarrow 4x + 14y = - 10 \\
\Rightarrow 2\left( {2x} \right) + 2\left( {7y} \right) = 2\left( { - 5} \right) \\
\Rightarrow 2x + 7y = - 5...................\left( {iv} \right) \\
 \]
Now we can see that (ii) and (iv) are equal, where $A = 2,\;B = 7\;,C = - 5$.

Therefore \[7x + 14y = 3x - 10\] in standard form can be written as \[2x + 7y = - 5\].

Note: PEMDAS is an acronym used to remind people of the order of operations. This means that you don't just solve math problems from left to right; rather, you solve them in a predetermined order that's given to you via the acronym PEMDAS.