The solution of an equation is, in fact, a numerical value that, if we place it in place of the unknown (or the variable), we will achieve equality between the two members of the equation, that is, we will obtain a "true statement". In first degree equations with one unknown, there can only be one solution.

Example:

X1=5X - 1 = 5

Solution of an equation x-1=5

This is an equation with one unknown or variable indicated by the letter XX. The equation is composed of two members separated by the use of the equal sign = = . The left member is everything to the left of the sign = = , and the right member is everything to the right of the sign.

Our goal is to isolate the variable (or clear the variable) X X so that only it remains in one of the members of the equation. In this way we will discover its value. In this article we will learn how to use the four mathematical operations(addition, subtraction, multiplication and division) to isolate the variable X X .

Practice Linear Equations

Examples with solutions for Linear Equations

Exercise #1

Solve the equation

5x15=30 5x-15=30

Video Solution

Step-by-Step Solution

We start by moving the sections:

5X-15 = 30
5X = 30+15

5X = 45

 

Now we divide by 5

X = 9

Answer

x=9 x=9

Exercise #2

Solve the equation

20:4x=5 20:4x=5

Video Solution

Step-by-Step Solution

To solve the exercise, we first rewrite the entire division as a fraction:

204x=5 \frac{20}{4x}=5

Actually, we didn't have to do this step, but it's more convenient for the rest of the process.

To get rid of the fraction, we multiply both sides of the equation by the denominator, 4X.

20=5*4X

20=20X

Now we can reduce both sides of the equation by 20 and we will arrive at the result of:

X=1

Answer

x=1 x=1

Exercise #3

Find the value of the parameter X

13x+56=16 \frac{1}{3}x+\frac{5}{6}=-\frac{1}{6}

Video Solution

Step-by-Step Solution

First, we will arrange the equation so that we have variables on one side and numbers on the other side.

Therefore, we will move 56 \frac{5}{6} to the other side, and we will get

13x=1656 \frac{1}{3}x=-\frac{1}{6}-\frac{5}{6}

Note that the two fractions on the right side share the same denominator, so you can subtract them:

 13x=66 \frac{1}{3}x=-\frac{6}{6}

Observe the minus sign on the right side!

 

13x=1 \frac{1}{3}x=-1

 

Now, we will try to get rid of the denominator, we will do this by multiplying the entire exercise by the denominator (that is, all terms on both sides of the equation):

1x=3 1x=-3

 x=3 x=-3

Answer

-3

Exercise #4

Solve the equation

413x=2123 4\frac{1}{3}\cdot x=21\frac{2}{3}

Video Solution

Step-by-Step Solution

We have an equation with a variable.

Usually, in these equations, we will be asked to find the value of the missing (X),

This is how we solve it:

 

To solve the exercise, first we have to change the mixed fractions to an improper fraction,

So that it will then be easier for us to solve them.

Let's start with the four and the third:

To convert a mixed fraction, we start by multiplying the whole number by the denominator

4*3=12

Now we add this to the existing numerator.

12+1=13

And we find that the first fraction is 13/3

 

Let's continue with the second fraction and do the same in it:
21*3=63

63+2=65

The second fraction is 65/3

We replace the new fractions we found in the equation:

 13/3x = 65/3

 

At this point, we will notice that all the fractions in the exercise share the same denominator, 3.

Therefore, we can multiply the entire equation by 3.

13x=65

 

Now we want to isolate the unknown, the x.

Therefore, we divide both sides of the equation by the unknown coefficient -
13.

 

63:13=5

x=5

Answer

x=5 x=5

Exercise #5

The area of the rectangle below is equal to 22x.

Calculate x.

x+8x+8x+8

Video Solution

Step-by-Step Solution

The area of the rectangle is equal to the length multiplied by the width.

Let's list the known data:

22x=12x×(x+8) 22x=\frac{1}{2}x\times(x+8)

22x=12x2+12x8 22x=\frac{1}{2}x^2+\frac{1}{2}x8

22x=12x2+4x 22x=\frac{1}{2}x^2+4x

0=12x2+4x22x 0=\frac{1}{2}x^2+4x-22x

0=12x218x 0=\frac{1}{2}x^2-18x

0=12x(x36) 0=\frac{1}{2}x(x-36)

For the equation to be equal, x needs to be equal to 36

Answer

x=36 x=36

Exercise #6

Given: the length of a rectangle is 3 greater than its width.

The area of the rectangle is equal to 27 cm².

Calculate the length of the rectangle

2727273x3x3xxxx

Video Solution

Step-by-Step Solution

The area of the rectangle is equal to length multiplied by width.

Let's set up the data in the formula:

27=3x×x 27=3x\times x

27=3x2 27=3x^2

273=3x23 \frac{27}{3}=\frac{3x^2}{3}

9=x2 9=x^2

x=9=3 x=\sqrt{9}=3

Answer

x=3 x=3

Exercise #7

(7x+3)×(10+4)=238 (7x+3)\times(10+4)=238

Video Solution

Step-by-Step Solution

We begin by solving the addition exercise in the right parenthesis:

(7x+3)+14=238 (7x+3)+14=238

We then multiply each of the terms inside of the parentheses by 14:

(14×7x)+(14×3)=238 (14\times7x)+(14\times3)=238

Following this we solve each of the exercises inside of the parentheses:

98x+42=238 98x+42=238

We move the sections whilst retaining the appropriate sign:

98x=23842 98x=238-42

98x=196 98x=196

Finally we divide the two parts by 98:

9898x=19698 \frac{98}{98}x=\frac{196}{98}

x=2 x=2

Answer

2

Exercise #8

(a+3a)×(5+2)=112 (a+3a)\times(5+2)=112

Calculate a a

Video Solution

Step-by-Step Solution

We begin by solving the two exercises inside of the parentheses:

4a×7=112 4a\times7=112

We then divide each of the sections by 4:

4a×74=1124 \frac{4a\times7}{4}=\frac{112}{4}

In the fraction on the left side we simplify by 4 and in the fraction on the right side we divide by 4:

a×7=28 a\times7=28

Remember that:

a×7=a7 a\times7=a7

Lastly we divide both sections by 7:

a77=287 \frac{a7}{7}=\frac{28}{7}

a=4 a=4

Answer

4

Exercise #9

4x:30=2 4x:30=2

Video Solution

Answer

x=15 x=15

Exercise #10

Solve the equation

7x+5.5=19.5 7x+5.5=19.5

Video Solution

Answer

x=2 x=2

Exercise #11

Solve the equation

8x10=80 8x\cdot10=80

Video Solution

Answer

x=1 x=1

Exercise #12

5x=0 5x=0

Video Solution

Answer

x=0 x=0

Exercise #13

5x=1 5x=1

What is the value of x?

Video Solution

Answer

x=15 x=\frac{1}{5}

Exercise #14

Find the value of the parameter X:

x+5=8 x+5=8

Video Solution

Answer

3

Exercise #15

Solve for X:

3+x=4 3+x=4

Video Solution

Answer

1