Solving Equations Using All Methods: Using fractions

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, $x = 4$.

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 $\frac{1}{3}$.

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):

$1x=-3$

 $x=-3$

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

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 $x = \frac{40}{27}$.

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 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 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 $\frac{9}{11} - \frac{8}{15}x = \frac{8}{22}$, 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 $\frac{75}{88}$.