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 $\frac{2}{8}x - 3 = 7$, 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 $\frac{1}{5}x - 4 = 6$, 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 $\frac{x}{4} + 2x - 18 = 0$, 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 $3b = \frac{7}{6}$ 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 $b = \frac{7}{18}$.

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 $x = 21\frac{1}{3}$.

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 solve the equation $\frac{x-4}{18} = \frac{7}{9}$, we'll follow these steps:

• Step 1: Apply the principle of cross-multiplication to eliminate fractions.

• Step 2: Solve for the linear expression in terms of $x$.

• Step 3: Isolate $x$ and solve the equation completely.

Now, let's work through each step:
Step 1: Cross-multiply to eliminate the fractions. The equation becomes:

$(x-4) \cdot 9 = 18 \cdot 7 \\ 9(x-4) = 126$

Step 2: Distribute the 9 on the left-hand side:

$9x - 36 = 126$

Step 3: Add 36 to both sides to isolate the term with $x$:

$9x = 126 + 36 9x = 162$

Step 4: Divide both sides by 9 to solve for $x$:

$x = \frac{162}{9} \\ x = 18$

Therefore, the solution to the equation is $x = 18$.

To solve the equation $\frac{x+2}{3}=\frac{4}{5}$, we can follow the method of cross-multiplication:

• Step 1: Cross-multiply to eliminate the fractions, giving us:

$(x + 2) \cdot 5 = 4 \cdot 3$

• Step 2: Simplify both sides of the equation:

$5(x + 2) = 12$

• Step 3: Distribute the 5 on the left side:

$5x + 10 = 12$

• Step 4: Subtract 10 from both sides to isolate the term with $x$:

$5x = 2$

• Step 5: Divide both sides by 5 to solve for $x$:

$x = \frac{2}{5}$

Therefore, the solution to the equation is $\frac{2}{5}$.

To solve for $x$ in the equation $\frac{7}{8}x = \frac{2}{5}$, we will follow these steps:

• Multiply both sides of the equation by the reciprocal of $\frac{7}{8}$, which is $\frac{8}{7}$.
• Simplify the resulting expression to find the value of $x$.

Let's work through these steps:

First, multiply both sides by $\frac{8}{7}$ to isolate $x$ on the left side.

$\frac{8}{7} \times \frac{7}{8}x = \frac{8}{7} \times \frac{2}{5}$

This simplifies to:

$x = \frac{8}{7} \times \frac{2}{5}$

Now, perform the multiplication of the fractions:

$x = \frac{8 \times 2}{7 \times 5} = \frac{16}{35}$

Thus, the value of $x$ is $\frac{16}{35}$.

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 $x = -3.73$.

We use the formula:

$\frac{a}{b}x=\frac{c}{d}$

$x=\frac{bc}{ad}$

We multiply the numerator by X and write the exercise as follows:

$\frac{x}{8}=\frac{3}{4}$

We multiply both sides by 8 to eliminate the fraction's denominator:

$8\times\frac{x}{8}=\frac{3}{4}\times8$

On the left side, it seems that the 8 is reduced and the right section is multiplied:

$x=\frac{24}{4}=6$

To solve the given linear equation, we will follow these steps:

• Step 1: Combine the terms involving $y$.
• Step 2: Simplify the constants on the right side of the equation.
• Step 3: Isolate $y$ to find its value.

Let’s solve the equation $\frac{1}{4}y + \frac{1}{2}y + 5 - 12 = 0$.

Step 1: Combine the like terms that involve $y$.
The coefficients of $y$ are $\frac{1}{4}$ and $\frac{1}{2}$. To combine them, we need a common denominator, which is 4. Therefore:

$\frac{1}{4}y + \frac{1}{2}y = \frac{1}{4}y + \frac{2}{4}y = \frac{3}{4}y$.

Step 2: Simplify the constants.
The equation now becomes $\frac{3}{4}y + 5 - 12 = 0$.
Combine the constants: $5 - 12 = -7$.

The equation simplifies to $\frac{3}{4}y - 7 = 0$.

Step 3: Isolate $y$.
Add 7 to both sides of the equation:
$\frac{3}{4}y = 7$.

To solve for $y$, multiply both sides by the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$:

$y = 7 \times \frac{4}{3} = \frac{28}{3}$.

Convert the fraction to a mixed number: $\frac{28}{3} = 9 \cdot 3 + 1 = 9 \text{ remainder } 1$. Thus, $\frac{28}{3} = 9\frac{1}{3}$.

Therefore, the value of $y$ is $9\frac{1}{3}$.

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 $x = 7$.

To solve the equation $17.5 - 18x - 5.5x = 19.2 + 14\frac{1}{2} - 5x$, follow these steps:

Step 1: Combine like terms on both sides of the equation.

• On the left side, combine $-18x - 5.5x$, which simplifies to $-23.5x$.
• On the right side, simplify $19.2 + 14.5 - 5x$. The fraction $14\frac{1}{2}$ is converted to decimal form as $14.5$, giving $19.2 + 14.5 = 33.7$.

Step 2: Rewrite the equation with the simplified terms:

$17.5 - 23.5x = 33.7 - 5x$.

Step 3: Get all terms involving $x$ on one side of the equation and constant terms on the other.

• Add $5x$ to both sides to move all $x$ related terms to the left:
• $17.5 - 23.5x + 5x = 33.7$

This further simplifies to $17.5 - 18.5x = 33.7$.

Step 4: Isolate the term with $x$ by subtracting $17.5$ from both sides:

$-18.5x = 33.7 - 17.5$.

The right side evaluates to $16.2$.

Thus, we have $-18.5x = 16.2$.

Step 5: Solve for $x$ by dividing both sides by $-18.5$:

$x = \frac{16.2}{-18.5} \approx -0.8757$.

Rounding $-0.8757$ to two decimal places gives $x = -0.87$.

Therefore, the solution to the equation is $x = -0.87$.

This corresponds to option 2 in the given choices.

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 this problem, we'll follow these steps:

• Step 1: Start with the given equation.
• Step 2: Use cross-multiplication to eliminate the fractions.
• Step 3: Simplify the resulting expression.
• Step 4: Solve for the variable $a$.

Now, let's work through each step:
Step 1: The equation given is $\frac{a}{6} = \frac{6}{7}$.
Step 2: We apply cross-multiplication: Multiply both sides to get $a \times 7 = 6 \times 6$.
Step 3: Simplify the equation: $7a = 36$.
Step 4: Solve for $a$ by dividing both sides by 7:
$a = \frac{36}{7}$.
This fraction can be converted to a mixed number: $a = 5\frac{1}{7}$.

Therefore, the solution to the problem is $a = 5\frac{1}{7}$.

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

• Step 1: Convert the mixed number to an improper fraction
• Step 2: Isolate $b$ using multiplication
• Step 3: Simplify to find the value of $b$

Now, let's work through each step:

Step 1: Convert $4\frac{1}{2}$ to an improper fraction:

$4\frac{1}{2} = \frac{9}{2}$

Step 2: Isolate $b$ on one side of the equation:

The equation becomes $70 = \frac{9}{2}b$

To isolate $b$, multiply both sides by the reciprocal of $\frac{9}{2}$:

$b = 70 \times \frac{2}{9}$

Step 3: Perform the multiplication:

$b = \frac{70 \times 2}{9}$

$b = \frac{140}{9}$

The improper fraction $\frac{140}{9}$ converts to a mixed number:

$b = 15 \frac{5}{9}$

Therefore, the solution to the problem is $b = 15\frac{5}{9}$.