Question

Solve the following equations:
1.\[20 = 6 + 2x\]
2.\[15 + x = 5x + 3\]
3.\[\dfrac{{3x + 2}}{{x - 6}} = - 7\]

Answer 1

Hint: Here we need to solve the given equation to find the value of \[x\] in each situation. We will solve these linear equations by applying mathematical operations on both sides of the equation. An equation is said to be a linear equation if the highest degree of a variable is equal to 1.

Complete step-by-step answer:
1.\[20 = 6 + 2x\]
We will subtract 6 from both the sides of the equation.
$\Rightarrow$ $20 - 6 = 6 + 2x - 6$
We will collect the like terms:
$\Rightarrow$ $14 = 6 - 6 + 2x$
$\Rightarrow$ $ 14 = 2x$
Dividing both sides by 2, we get
$\Rightarrow$ $\dfrac{{14}}{2} = \dfrac{{2x}}{2}$
$\Rightarrow$ $x=7$
The value of \[x\] is 7.

2.\[15 + x = 5x + 3\]
We will subtract 3 from both the sides of the equation.
\[15 + x - 3 = 5x + 3 - 3\]
We will collect the like terms:
$\Rightarrow$ $15 - 3 + x = 5x + 3 - 3$
$\Rightarrow$ $12 + x = 5x$
Subtracting both side by \[x\], we get
$\Rightarrow$ $12 + x - x = 5x - x$
$\Rightarrow$ $12 = 4x$
Dividing both side by 4,we get
$\Rightarrow$ $\dfrac{{12}}{4} = \dfrac{{4x}}{4}$
$\Rightarrow$ $3 = x$
Thus, the value of \[x\] is 3.

3.\[\dfrac{{3x + 2}}{{x - 6}} = - 7\]
Multiplying both sides by \[x - 6\], we get
\[\dfrac{{3x + 2}}{{x - 6}}\left( {x - 6} \right) = - 7\left( {x - 6} \right)\]
Apply distributive property to the question, we get
$\Rightarrow$ $3x + 2 = - 7 .(x) + (- 7).( - 6)$
$\Rightarrow$ $3x + 2 = - 7x + 42$
Subtracting 2 from both sides, we get
$\Rightarrow$ $3x + 2 - 2 = - 7x + 42 - 2$
$\Rightarrow$ $3x = - 7x + 40$
Adding \[7x\] from both sides, we get
$\Rightarrow$ $3x + 7x = - 7x + 40 + 7x$
$\Rightarrow$ $3x + 7x= - 7x + 7x+ 40$
$\Rightarrow$ $10x = 40$
Dividing, both side by 10, we get
$\Rightarrow$ \[\dfrac{{10x}}{{10}} = \dfrac{{40}}{{10}}\\
$\Rightarrow$ x = 4
\]
The value of \[x\] is 4.

Note: We might feel confused about which operation should be applied on the linear equation. We should keep in mind that we have to find the value of the variable \[x\]. If some number is added to \[x\] in the equation, we should subtract that number from both sides of the equation. \[x\]. If some number is subtracted from \[x\] in the equation, we should add that number to both sides of the equation. If \[x\] is multiplied to a number in the equation, we should divide both sides of the equation by that number. Similarly, if \[x\] is divided by a number in the equation, we should multiply both sides of the equation by that number to solve the equation.
We have also used distributive property to simplify the equation. According to the distributive property multiplying the sum of two or more numbers by a number will give the same result as multiplying each number individually by the number and then adding the products: \[a\left( {b + c} \right) = ab + ac\].