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

Solve for X:

6x=102 6 - x = 10 - 2

Step-by-Step Solution

To solve the equation 6x=102 6 - x = 10 - 2 , follow these steps:

  1. First, simplify both sides of the equation:

  2. On the right side, calculate 102=8 10 - 2 = 8 .

  3. The equation simplifies to 6x=8 6 - x = 8 .

  4. To isolate x, subtract 6 from both sides:

  5. 6x6=86 6 - x - 6 = 8 - 6

  6. This simplifies to x=2 -x = 2 .

  7. Multiply both sides by -1 to solve for x:

  8. x=2×1=2 x = -2 \times -1 = 2 .

  9. Since the problem requires only manipulation by transferring terms, the initial approach to the equation setup should lead to x = 4 as the solution before re-evaluation.

Therefore, the correct solution to the equation is x=2 x=2 .

Answer

2

Exercise #4

Solve for X:

5x=124 5 - x = 12 - 4

Step-by-Step Solution

First, simplify the right side of the equation:
124=8 12 - 4 = 8
Hence, the equation becomes 5x=8 5 - x = 8 .
Subtract 5 from both sides to isolate x x :
5x5=85 5 - x - 5 = 8 - 5
This simplifies to:
x=3 -x=3
Divide by -1 to solve for x x :
x=3 x=-3
Therefore, the solution is x=3 x=-3 .

Answer

-3

Exercise #5

Solve for X:

7x=155 7 - x = 15 - 5

Step-by-Step Solution

First, simplify the right side of the equation:
155=10 15 - 5 = 10
Hence, the equation becomes 7x=10 7 - x = 10 .
Subtract 7 from both sides to isolate x x :
7x7=107 7 - x - 7 = 10 - 7
This simplifies to:
x=3 -x=3
Divide by -1 to solve forx x :
x=3 x=-3
Therefore, the solution is x=3 x=-3 .

Answer

-3

Exercise #6

Solve for X:

9x=167 9 - x = 16 - 7

Step-by-Step Solution

First, simplify the right side of the equation:
167=9 16 - 7 = 9
Hence, the equation becomes 9x=9 9 - x = 9 .
Since both sides are equal, x x must be 0 0 .
Therefore, the solution is x=0 x = 0 .

Answer

0

Exercise #7

Solve for X:

3x=106 3 - x = 10 - 6

Step-by-Step Solution

First, simplify the right side of the equation:
106=4 10 - 6 = 4
Hence, the equation becomes 3x=4 3 - x = 4 .
Subtract 3 from both sides to isolate x x :
3x3=43 3 - x - 3 = 4 - 3
This simplifies to:
x=1 -x=1
Divide by -1 to solve forx x :
x=1 x=-1
Therefore, the solution is x=1 x = 1 .

Answer

-1

Exercise #8

Solve for X:

3+x2=73 3 + x - 2 = 7 - 3

Step-by-Step Solution

First, simplify both sides of the equation:

Left side: 3+x2=1+x 3 + x - 2 = 1 + x

Right side: 73=4 7 - 3 = 4

So the equation becomes:

1+x=4 1 + x = 4

Next, isolate x x by subtracting 1 from both sides:

1+x1=41 1 + x - 1 = 4 - 1

This simplifies to:

x=3 x = 3

Answer

3

Exercise #9

Solve for X:

5+x3=2+1 5 + x - 3 = 2 + 1

Step-by-Step Solution

To solve 5+x3=2+1 5 + x - 3 = 2 + 1 , we first simplify both sides:

Left side:
53+x=2+x 5 - 3 + x = 2 + x

Right side:
2+1=3 2 + 1 = 3

Now the equation is 2+x=3 2 + x = 3 .

Subtract 2 from both sides:
x=32 x = 3 - 2

So, x=1 x = 1 .

Answer

1

Exercise #10

Solve for X:

3+x+1=62 3 + x + 1 = 6 - 2

Step-by-Step Solution

To solve 3+x+1=62 3 + x + 1 = 6 - 2 , we first simplify both sides:

Left side:
3+1+x=4+x 3 + 1 + x = 4 + x

Right side:
62=4 6 - 2 = 4

Now the equation is 4+x=4 4 + x = 4 .

Subtract 4 from both sides:
x=44 x = 4 - 4

So, x=0 x = 0 .

Answer

0

Exercise #11

Solve for X:

x+42=6+1 x + 4 - 2 = 6 + 1

Step-by-Step Solution

First, simplify both sides of the equation:

Left side: x+42=x+2 x + 4 - 2 = x + 2

Right side: 6+1=7 6 + 1 = 7

Now the equation is: x+2=7 x + 2 = 7

Subtract 2 from both sides to isolatex x :

x+22=72 x + 2 - 2 = 7 - 2

Simplifying gives:

x=5 x = 5

Answer

5

Exercise #12

Solve for X:

x3+5=82 x - 3 + 5 = 8 - 2

Step-by-Step Solution

First, simplify both sides of the equation:

Left side: x3+5=x+2 x - 3 + 5 = x + 2

Right side: 82=6 8 - 2 = 6

Now the equation is: x+2=6 x + 2 = 6

Subtract 2 from both sides to isolate x x :

x+22=62 x + 2 - 2 = 6 - 2

Simplifying gives:

x=4 x = 4

Answer

4

Exercise #13

Solve for x x :

5x3=45 5x \cdot 3 = 45

Step-by-Step Solution

To solve the equation5x3=45 5x \cdot 3 = 45 , follow these steps:

1. First, identify the operation needed to solve forx x . In this case, we have a multiplication equation.

2. Therefore, we divide both sides of the equation by 15 (since 5×3=15 5 \times 3 = 15 ) to isolate x x :

x=4515 x = \frac{45}{15}

3. Calculate x x :

x=3 x = 3

Answer

x=3 x=3

Exercise #14

Solve the equation:

6x2=24 6x \cdot 2 = 24

Step-by-Step Solution

To solve the equation 6x2=24 6x \cdot 2 = 24 , follow these steps:

1. First, identify the operation involved. In this case, it is multiplication.

2. Divide both sides of the equation by 12 (since 6×2=12 6 \times 2 = 12 ) to isolate x x :

x=2412 x = \frac{24}{12}

3. Calculate x x :

x=2 x = 2

Answer

x=2 x=2

Exercise #15

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