Solving an Equation by Multiplication/ Division: Using fractions

To solve this problem, we'll follow these steps:

• Identify the given fraction equation.
• Multiply both sides of the equation by the reciprocal of the coefficient of $x$.
• Simplify to isolate $x$.

Now, let's work through these steps:
Step 1: The problem gives us the equation $\frac{1}{3} x = \frac{1}{9}$.
Step 2: We multiply both sides by 3 to eliminate the fraction on the left side:

$3 \times \frac{1}{3} x = 3 \times \frac{1}{9}$

Step 3: Simplifying both sides results in:

$x = \frac{3}{9}$

Further simplification of $\frac{3}{9}$ yields:

$x = \frac{1}{3}$

Therefore, the solution to the problem is $\frac{1}{3}$.

We begin by multiplying the simple fraction by y:

$\frac{-1}{5}\times y=-25$

We then reduce both terms by $-\frac{1}{5}$

$y=\frac{-25}{-\frac{1}{5}}$

Finally we multiply the fraction by negative 5:

$y=-25\times(-5)=125$

To solve the equation $3\frac{1}{2} \cdot y = 21$, we'll follow these steps:

• Convert the mixed number to an improper fraction.
• Divide both sides of the equation by the coefficient of $y$.

Let's analyze these steps in detail:

Step 1: Convert the mixed number to an improper fraction.
The coefficient of $y$ is $3\frac{1}{2}$. Converting to an improper fraction, we have:

$3\frac{1}{2} = \frac{7}{2}$

Step 2: Divide both sides of the equation by $\frac{7}{2}$.
The equation becomes:

$\frac{7}{2} \cdot y = 21$

To isolate $y$, divide both sides by $\frac{7}{2}$:

$y = 21 \div \frac{7}{2}$

Dividing by a fraction is equivalent to multiplying by its reciprocal, so:

$y = 21 \cdot \frac{2}{7}$

Carrying out the multiplication, we calculate:

$y = \frac{21 \cdot 2}{7} = \frac{42}{7}$

Dividing the numerator by the denominator gives us:

$y = 6$

Thus, the solution to the equation is $y = 6$.

To solve the equation $\frac{1}{5}x - 4 = 6$, we will follow these steps:

• Step 1: Add 4 to both sides of the equation to eliminate the subtraction and isolate the fractional term.
• Step 2: Multiply both sides by 5 to clear the fraction and solve for $x$.

Let's apply these steps to solve the equation:

Step 1: Add 4 to both sides:
$\frac{1}{5}x - 4 + 4 = 6 + 4$
This simplifies to:
$\frac{1}{5}x = 10$

Step 2: Multiply both sides by 5 to solve for $x$:
$5 \times \frac{1}{5}x = 10 \times 5$
This simplifies to:
$x = 50$

Therefore, the solution to the equation is $x = 50$.

To solve the equation $\frac{2}{8}x - 3 = 7$, we'll follow these steps:

• Step 1: Simplify the fraction. The coefficient $\frac{2}{8}$ simplifies to $\frac{1}{4}$.
• Step 2: Eliminate the constant term by adding 3 to both sides of the equation.
• Step 3: Solve for $x$ by removing the coefficient of $x$ using division.

Let's solve the equation step-by-step:

Step 1: Simplify the equation:
The equation $\frac{2}{8}x - 3 = 7$ simplifies to $\frac{1}{4}x - 3 = 7$.

Step 2: Eliminate the constant term:
Add 3 to both sides to isolate the term involving $x$:

$\frac{1}{4}x - 3 + 3 = 7 + 3$

This simplifies to:

$\frac{1}{4}x = 10$

Step 3: Solve for $x$:
Multiply both sides by the reciprocal of $\frac{1}{4}$ to solve for $x$:

$4 \cdot \frac{1}{4}x = 4 \cdot 10$

This simplifies to:

$x = 40$

Therefore, the solution to the equation is $x = 40$.

To solve the equation $\frac{x}{4} + 2x - 18 = 0$, we proceed as follows:

• Step 1: Eliminate the fraction by multiplying the entire equation by 4:
  $(4) \Big(\frac{x}{4} + 2x - 18\Big) = (4)(0)$
• Step 2: Distribute and simplify:
  $x + 8x - 72 = 0$
• Step 3: Combine like terms:
  $9x - 72 = 0$
• Step 4: Isolate $9x$ by adding 72 to both sides:
  $9x = 72$
• Step 5: Solve for $x$ by dividing both sides by 9:
  $x = \frac{72}{9}$
• Step 6: Simplify the division:
  $x = 8$

Thus, the solution to the problem is $x = 8$.

To solve the equation $\frac{3x}{4} = 16$, we will eliminate the fraction by multiplying both sides by 4.

• Step 1: Multiply both sides by 4:
  $\left(\frac{3x}{4}\right) \times 4 = 16 \times 4$
• Step 2: Simplify:
  $3x = 64$
• Step 3: Solve for $x$ by dividing both sides by 3:
  $x = \frac{64}{3}$
• Step 4: Simplify the fraction to a mixed number:
  $x = 21\frac{1}{3}$

Therefore, the solution to the equation $\frac{3x}{4} = 16$ is $x = 21\frac{1}{3}$.

To solve the equation $3b = \frac{7}{6}$ for the variable $b$, we will perform the following steps:

• Step 1: Identify the equation. The given equation is $3b = \frac{7}{6}$.
• Step 2: Isolate the variable. Divide both sides by 3 to solve for $b$.

When we divide both sides of the equation by 3, we obtain:

Step 3: Simplify the expression. Dividing a fraction by an integer is equivalent to multiplying the denominator of the fraction by that integer:

The denominator becomes:

Thus, the solution to the equation is $b = \frac{7}{18}$.

This matches the correct answer choice among the given options.

Therefore, the value of $b$ is $b = \frac{7}{18}$.

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

To solve this linear equation, we'll take the following steps:

• Step 1: Multiply both sides of the equation by 2 to eliminate the fraction.
• Step 2: Simplify and isolate the term containing $x$.
• Step 3: Solve for $x$ by further isolation.

Let's execute these steps:

Step 1: Start with the given equation:

$\frac{-5 + 7x}{2} = 22$

Multiply both sides by 2 to remove the fraction:

$-5 + 7x = 44$

Step 2: Now, eliminate the constant term on the left side by adding 5 to both sides:

$-5 + 7x + 5 = 44 + 5$

This simplifies to:

$7x = 49$

Step 3: Finally, solve for $x$ by dividing both sides by 7:

$x = \frac{49}{7}$

Calculate the result:

$x = 7$

Therefore, the value of $x$ is $x = 7$.

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

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

$\frac{1}{3}x=-\frac{1}{6}-\frac{5}{6}$

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

$\frac{1}{3}x=-\frac{6}{6}$

Observe the minus sign on the right side!

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

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):

$1x=-3$

$x=-3$

To find the value of $x$ in the given equation, we will perform the following steps:

• Step 1: Start with the equation given: $3x - \frac{1}{9} = \frac{8}{9}$.
• Step 2: To eliminate the constant $-\frac{1}{9}$ on the left, add $\frac{1}{9}$ to both sides:

$3x - \frac{1}{9} + \frac{1}{9} = \frac{8}{9} + \frac{1}{9}$

This simplifies to:

$3x = \frac{8}{9} + \frac{1}{9}$

Combine the fractions on the right side:

$\frac{8}{9} + \frac{1}{9} = \frac{9}{9} = 1$

So, now we have:

$3x = 1$

• Step 3: Divide both sides by 3 to solve for $x$:

$x = \frac{1}{3}$

Thus, the solution to the equation is:

$x = \frac{1}{3}$

To solve the equation $\frac{1}{6}x - \frac{1}{3} = \frac{1}{3}$, we will take the following steps:

• Step 1: Eliminate fractions by multiplying the entire equation by the least common multiple of the denominators $6$.
• Step 2: Simplify the resulting equation.
• Step 3: Isolate the variable $x$.

Let's proceed with the solution:

Step 1: Multiply the entire equation by $6$ to clear fractions:
$6 \left(\frac{1}{6}x - \frac{1}{3}\right) = 6 \times \frac{1}{3}$

Step 2: Simplify:
$x - 2 = 2$

Step 3: Solve for $x$ by adding $2$ to both sides:
$x = 2 + 2$

Therefore, $x = 4$.

Let's solve the equation $\frac{2}{3}x - \frac{4}{6} = \frac{1}{3}$.

Step 1: Simplify the fractions.

• The term $\frac{4}{6}$ is equivalent to $\frac{2}{3}$ after simplification.

Now, the equation can be rewritten as:

$\frac{2}{3}x - \frac{2}{3} = \frac{1}{3}$

Step 2: Add $\frac{2}{3}$ to both sides to isolate the term with $x$.

$\frac{2}{3}x = \frac{1}{3} + \frac{2}{3}$

Simplify the right side:

$\frac{2}{3}x = \frac{3}{3}$

$\frac{3}{3} = 1$

So the equation becomes:

$\frac{2}{3}x = 1$

Step 3: Solve for $x$ by multiplying both sides by the reciprocal of $\frac{2}{3}$.

Multiply both sides by $\frac{3}{2}$:

$x = 1 \times \frac{3}{2}$

Thus, the solution is:

$x = \frac{3}{2}$

The solution to the problem is $x = \frac{3}{2}$.

Let's proceed with solving the equation step by step:

1. Start with the equation $\frac{2}{3}x + \frac{1}{4} = \frac{3}{4}$.

2. Subtract $\frac{1}{4}$ from both sides to remove the constant term on the left:
   $\frac{2}{3}x + \frac{1}{4} - \frac{1}{4} = \frac{3}{4} - \frac{1}{4}$.

3. This simplifies to: $\frac{2}{3}x = \frac{3}{4} - \frac{1}{4}$.

4. Perform the subtraction on the right-hand side:
   $\frac{2}{3}x = \frac{2}{4} = \frac{1}{2}$.

5. Now solve for $x$ by dividing both sides of the equation by $\frac{2}{3}$:
   $x = \frac{1}{2} \div \frac{2}{3}$.

6. Dividing by a fraction is the same as multiplying by its reciprocal:
   $x = \frac{1}{2} \times \frac{3}{2}$.

7. Simplify the multiplication:
   $x = \frac{3}{4}$.

Therefore, the value of the parameter $x$ is $\frac{3}{4}$.

To solve the linear equation $\frac{4}{5}x + \frac{3}{7} = \frac{2}{14}$, we will follow these steps:

• Step 1: Subtract $\frac{3}{7}$ from both sides of the equation to isolate the term with $x$.
• Step 2: Simplify the resulting equation.
• Step 3: Solve for $x$ by multiplying both sides by the reciprocal of the coefficient of $x$.

Now, let's work through the solution:

Step 1: Subtract $\frac{3}{7}$ from both sides:

$\frac{4}{5}x = \frac{2}{14} - \frac{3}{7}$

Step 2: Simplify the right side:

$\frac{2}{14}$ can be simplified to $\frac{1}{7}$, so the equation becomes:

$\frac{4}{5}x = \frac{1}{7} - \frac{3}{7}$

Simplifying the right side gives:

$\frac{4}{5}x = -\frac{2}{7}$

Step 3: Solve for $x$.

Multiply both sides by the reciprocal of $\frac{4}{5}$, which is $\frac{5}{4}$:

$x = -\frac{2}{7} \times \frac{5}{4}$

Perform the multiplication on the right side:

$x = -\frac{2 \times 5}{7 \times 4} = -\frac{10}{28}$

Simplify $-\frac{10}{28}$ by dividing the numerator and the denominator by their greatest common divisor, which is 2:

$x = -\frac{5}{14}$

Thus, the solution to the equation is $x = -\frac{5}{14}$.

To solve the equation $\frac{4}{9} + \frac{3}{5}x = \frac{4}{3}$, we will follow these steps:

• Step 1: Subtract $\frac{4}{9}$ from both sides to isolate the term involving $x$.
• Step 2: Divide by the coefficient of $x$ to solve for $x$.

Step 1: Subtract $\frac{4}{9}$ from both sides:

$\frac{3}{5}x = \frac{4}{3} - \frac{4}{9}$

To subtract these fractions, find a common denominator. The least common denominator for 3 and 9 is 9. Rewrite $\frac{4}{3}$ as $\frac{12}{9}$ (since $4 \times 3 = 12$), resulting in:

$\frac{3}{5}x = \frac{12}{9} - \frac{4}{9} = \frac{8}{9}$

Step 2: Divide both sides by $\frac{3}{5}$ to solve for $x$:

$x = \frac{8}{9} \div \frac{3}{5} = \frac{8}{9} \times \frac{5}{3}$

Multiply the fractions. The result is:

$x = \frac{8 \times 5}{9 \times 3} = \frac{40}{27}$

Thus, the solution to the equation is $x = \frac{40}{27}$.

To solve the equation $\frac{9}{11} - \frac{8}{15}x = \frac{8}{22}$, follow these steps:

• Step 1: Find the least common denominator (LCD) of 11, 15, and 22, which is 330.
• Step 2: Multiply every term in the equation by 330 to eliminate the fractions:
  $330 \times \frac{9}{11} - 330 \times \frac{8}{15}x = 330 \times \frac{8}{22}$.
• Step 3: Simplify each term:
  - For $\frac{9}{11}$: $330 \times \frac{9}{11} = 270$,
  - For $\frac{8}{15}x$: $330 \times \frac{8}{15}x = 176x$,
  - For $\frac{8}{22}$: $330 \times \frac{8}{22} = 120$.
• Step 4: Substitute back into the equation:
  $270 - 176x = 120$.
• Step 5: Isolate $x$:
  - Subtract 270 from both sides: $-176x = 120 - 270$,
  - Simplify: $-176x = -150$,
  - Divide both sides by $-176$: $x = \frac{-150}{-176} = \frac{75}{88}$.

Thus, the value of $x$ is $\frac{75}{88}$.

To solve the equation $\frac{8}{3} - \frac{4}{5}x = -\frac{2}{10}x$, follow these steps:

• Step 1: Identify the least common denominator (LCD) of the fractions involved. The denominators are 3, 5, and 10, so the LCD is 30.
• Step 2: Multiply the entire equation by 30 to eliminate the fractions:
  $30 \times \left(\frac{8}{3} - \frac{4}{5}x\right) = 30 \times \left(-\frac{2}{10}x\right)$
• Step 3: Simplify each term:
  For $\frac{8}{3}$: $30 \times \frac{8}{3} = 10 \times 8 = 80$
  For $\frac{4}{5}x$: $30 \times \frac{4}{5}x = 6 \times 4x = 24x$
  For $-\frac{2}{10}x$: $30 \times -\frac{2}{10}x = 3 \times -2x = -6x$
• Step 4: Rewrite the equation:
  $80 - 24x = -6x$
• Step 5: Combine like terms by moving terms containing $x$ to one side:
  Subtract $-6x$ from both sides:
  $80 = 18x$
• Step 6: Solve for $x$ by dividing both sides by 18:
  $x = \frac{80}{18} = \frac{40}{9}$ after simplification.

Therefore, the solution to the problem is $x = \frac{40}{9}$.

To solve the equation $22x - \frac{1}{2} + 16\frac{1}{2} = 14.5x - 12$, we will follow these steps:
1. Combine like terms on both sides of the equation.
2. Isolate the variable $x$.
3. Solve for $x$.

Let's start by simplifying each side:

• Simplify the left-hand side: $22x - \frac{1}{2} + 16\frac{1}{2}$.

The term $16\frac{1}{2}$ is equivalent to $16.5$, so the left-hand side becomes:
$22x - 0.5 + 16.5 = 22x + 16$.

• Now simplify the right-hand side: $14.5x - 12$.

The right-hand side remains as $14.5x - 12$.

Now, let's collect like terms. Move the term involving $x$ from the right-hand side to the left:

• Subtract $14.5x$ from both sides:
  $22x + 16 - 14.5x = -12$

This simplifies to:
$7.5x + 16 = -12$.

Next, isolate the constant term. Subtract 16 from both sides:

• $7.5x + 16 - 16 = -12 - 16$

This simplifies to:
$7.5x = -28$.

Finally, solve for $x$ by dividing both sides by 7.5:

• $x = \frac{-28}{7.5}$

Calculating the fraction gives approximately:
$x \approx -3.73$.

Therefore, the solution to the problem is $x = -3.73$.