Question

How do you write \[y - 4{\text{ }} = {\text{ }}2.5\left( {x + 3} \right)\] in standard form?

Answer 1

Hint: In this question, we have been given a linear equation which we have to convert in standard form. The standard form of a linear equation is: \[Ax + By = C\] where, if at all possible, A, B, and C are integers, and A is non-negative, and, A, B, and C have no common factors other than 1.

Complete step-by-step solution:
In the given question, first, we must multiply each side of the equation by 2 to ensure all coefficients are integers as shown below:
\[2\left( {y - 4} \right) = 2 \times 2.5\left( {x + 3} \right)\] ………….. (1)
Simplifying equation (1) we will get a further simplified form of our equation that is shown as follows:
$\Rightarrow$\[2y - 8 = 5\left( {x + 3} \right)\;\] ………….... (2)
Next, expand the terms on the right side of the equation by multiplying each term within the brackets by the term outside the brackets:
$\Rightarrow$\[2y - 8 = \left( {5 \times x} \right) + \left( {5 \times 3} \right)\]
Finally, we get the most simplified form of our equation as given below:
$\Rightarrow$\[2y - 8 = 5x + 15\] …………….. (3)
Then, add 8 and subtract 5x from each side of the equation (3) to place the x and y terms on the left side of the equation and constant on the right side of the equation while keeping the equation balanced:
$\Rightarrow$\[ - 5x + 2y - 8 + 8 = - 5x + 5x + 15 + 8\].................(4)
Upon simplification after performing operations in equation (4), we will get our equation (5):
$\Rightarrow$\[ - 5x + 2y - 0 = 0 + 23\]
$\Rightarrow$\[ - 5x + 2y = 23\].....................(5)
Now, multiply each side of the equation (5) by −1 to ensure the x coefficient is positive while keeping the equation balanced:
$\Rightarrow$\[ - 1\left( { - 5x + 2y} \right) = - 1 \times 23\]
Multiplying -1 within the brackets for both the terms to get our further simplification as shown below:
$\Rightarrow$\[\left( { - 1 \times - 5x} \right) + \left( { - 1 \times 2y} \right) = - 23\]
Thus, the equation in our standard form can be written as shown below:
\[5x - 2y = - 23\]

Thus, our final standard form equation is \[5x - 2y = - 23\] .

Note: Some prefer that the lead coefficient A must be a positive number. In case if it is not positive then we can also multiply every term by (-1) and then simplify our equations.
Other major methods or forms of linear equations are: point-slope form and slope-intercept form. So, we can also derive the results using these approaches.