Question

How do you write $y - 3 = - 2.4(x - 5)$ in standard form?

Answer 1

Hint:
In this question, we need to write the given expression in the standard form. Firstly, we will multiply $ - 2.4$ to $(x - 5)$ using the distributive property given by $a \cdot (b - c) = a \cdot b - b \cdot c$. We then try to write it in a standard form of a linear equation which is given by, $Ax + By = C$. Hence, we make rearrangement in the given equation and try to convert it in the general form. After that, we will get the solution in the standard form which is the required answer.

Complete step by step solution:
Let us solve the question.
Given an equation of the form, $y - 3 = - 2.4(x - 5)$ …… (1)
We are asked to find the standard form of the above expression given in the equation (1).
Observe the given equation carefully.
Firstly, we will simplify the R.H.S. of the equation (1).
Consider the R.H.S. given by $ - 2.4(x - 5)$
We multiply this using distributive property given by,
$a \cdot (b - c) = a \cdot b - b \cdot c$
Here $a = - 2.4,$ $b = x,$ $c = 5$
Hence we get,
$ \Rightarrow - 2.4(x - 5) = - 2.4 \times x - ( - 2.4) \times 5$
$ \Rightarrow - 2.4(x - 5) = - 2.4x + 12$
Hence the equation (1) becomes,
$ \Rightarrow y - 3 = - 2.4x + 12$ …… (2)
Now let us convert the above equation into standard form.
We know that the standard form of a linear equation is given by,
$Ax + By + C = 0$
Where, if at all possible A, B and C are integers and A is non negative.
Also A, B and C have no common factor other than 1.
Hence, we convert the given equation into standard form as follows.
From equation (2), we have,
$y - 3 = - 2.4x + 12$
Adding 3 to both sides of the equation we get,
$ \Rightarrow y - 3 + 3 = - 2.4x + 12 + 3$
Combining like terms $ - 3 + 3 = 0$
Combining like terms $12 + 3 = 15$
Hence we get,
$ \Rightarrow y + 0 = - 2.4x + 15$
$ \Rightarrow y = - 2.4x + 15$
Adding $2.4x$ on both sides we get,
$ \Rightarrow y + 2.4x = 2.4x - 2.4x + 15$
Combining like terms $2.4x - 2.4x = 0$
Hence we get,
$ \Rightarrow y + 2.4x = 0 + 15$
$ \Rightarrow y + 2.4x = 15$
Now multiply the each side of the above equation by 5, to convert all the coefficients to integers which keeps the equation balanced.
Multiplying by 5 on both sides we get,
$ \Rightarrow 5(y + 2.4x) = 5 \times 15$
By using distributive property again we get,
$ \Rightarrow 5.y + 5 \cdot (2.4x) = 75$
$ \Rightarrow 5y + 12x = 75$
Rearranging the above equation we get,
$ \Rightarrow 12x + 5y = 75$
Which is in the standard form of a linear equation.

Hence, the standard form of an equation $y - 3 = - 2.4(x - 5)$ is given by $12x + 5y = 75$.

Note:
Student must remember the standard form of a linear equation which is given as,
$Ax + By + C = 0$
Where, if at all possible A, B and C are integers and A is non negative.
Also A, B and C have no common factor other than 1.
To get an equation into standard form, follow the steps given below.
(1) Isolate the constant term i.e. the term with no variable on the right hand side of the equation by just adding and subtracting terms from both sides.
(2) If there are any fractions involved, multiply the whole equation by the lowest common denominator.
(3) If the coefficient of A is negative, then multiply the whole equation by -1.