Solving an Equation by Multiplication/ Division: Equations with variables on both sides

We use the formula:

$a\cdot x=b$

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

Note that the coefficient of X is 3.

Therefore, we will divide both sides by 3:

$\frac{3x}{3}=\frac{18}{3}$

Then divide accordingly:

$x=6$

To solve the equation $-7y = -27$, we need to isolate the variable $y$. We do this by performing the following steps:

• Step 1: Divide both sides of the equation by $-7$ to solve for $y$.

Performing this operation gives us:

Simplifying both sides, we have:

To express $\frac{27}{7}$ as a mixed number, we divide 27 by 7:

• 27 divided by 7 equals 3 with a remainder of 6. Hence, $\frac{27}{7} = 3\frac{6}{7}$.

Therefore, the solution to the equation is $y = 3\frac{6}{7}$.

Among the given choices, option 1 matches our result.

Therefore, the solution to the problem is $y = 3\frac{6}{7}$.

To solve the equation $8x = 5$, follow these steps:

• Step 1: Identify the equation $8x = 5$, where $x$ is the unknown variable.
• Step 2: To isolate $x$, divide both sides of the equation by 8.
  This step involves equivalent operations to maintain equality.
• Step 3: Perform the division on both sides:
  $\frac{8x}{8} = \frac{5}{8}$.
  This simplifies to $x = \frac{5}{8}$.

Now, let's outline these steps in detail:

We begin with the equation $8x = 5$.

Dividing both sides by the coefficient of $x$, which is 8, gives:

$\frac{8x}{8} = \frac{5}{8}$.

This simplifies directly to:

$x = \frac{5}{8}$.

Therefore, the solution to the problem is $x = \frac{5}{8}$.

To solve the equation $5x = 3$, we will isolate $x$ by using division:

• Step 1: Recognize that $x$ is multiplied by 5. To isolate $x$, we need to undo this multiplication.
• Step 2: Divide both sides of the equation by 5. This step uses the Division Property of Equality:

$\frac{5x}{5} = \frac{3}{5}$

Step 3: Simplify both sides. The left side simplifies to $x$ (because $\frac{5x}{5} = x$), and the right side is $\frac{3}{5}$.

Hence, the solution to the equation $5x = 3$ is $x = \frac{3}{5}$.

To solve the equation $7x = 4$, we will follow these steps:

• Step 1: We start with the equation $7x = 4$.

• Step 2: Our goal is to isolate $x$. Since $x$ is multiplied by 7, we will divide both sides of the equation by 7.

• Step 3: Performing division: $x = \frac{4}{7}$

Therefore, the solution to the equation $7x = 4$ is $x = \frac{4}{7}$.

We use the formula:

$a\cdot x=b$

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

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

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

We multiply by 4 to get rid of the fraction's denominator:

$4\times\frac{x}{4}=3\times4$

Then, we remove the common factor from the left side and perform the multiplication on right side to obtain:

$x=12$

To solve the equation $-6x = 18$, we need to isolate the variable $x$.

Our equation is:

$-6x = 18$

The variable $x$ is multiplied by $-6$. To undo this operation and solve for $x$, we divide both sides of the equation by $-6$. This will isolate $x$ on one side of the equation:

$\frac{-6x}{-6} = \frac{18}{-6}$

Simplifying both sides, we find:

$x = -3$

Thus, the solution to the equation $-6x = 18$ is $x = -3$.

Therefore, the correct answer is $x = -3$.

To solve the equation $6x = 3$, follow these steps:

Step 1: We aim to isolate $x$. Divide both sides of the equation by 6 to remove the coefficient attached to $x$:

Step 2: Simplify the fraction on the right side:

Therefore, the solution to the equation $6x = 3$ is $x = \frac{1}{2}$.

To solve the equation $4x = \frac{1}{8}$, we need to isolate $x$. We do this by dividing both sides of the equation by the coefficient of $x$, which is 4:

• Step 1: Write the original equation: $4x = \frac{1}{8}$.
• Step 2: Divide both sides by 4 to solve for $x$:

$x = \frac{\frac{1}{8}}{4}$

• Step 3: Simplify the right-hand side by multiplying fractions, recalling that dividing by a number is equivalent to multiplying by its reciprocal:

$x = \frac{1}{8} \times \frac{1}{4} = \frac{1 \times 1}{8 \times 4} = \frac{1}{32}$

Thus, the solution to the equation is $x = \frac{1}{32}$.

$ax=\frac{c}{b}$

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

To solve for$x$, first, get all terms involving $x$ on one side and constants on the other. Start from:

$4x - 7 = x + 5$

Subtract $x$ from both sides to simplify:

$3x - 7 = 5$

Add 7 to both sides to isolate the terms with$x$:

$3x = 12$

Divide each side by 3 to solve for$x$:

$x = 4$

Thus, $x$ is $4$.

To solve this linear equation, follow these steps:

• Step 1: Isolate the term with $x$ by subtracting 7 from both sides of the equation.
• Step 2: Perform the subtraction to simplify the equation.
• Step 3: Divide both sides by 5 to solve for $x$.

Let’s perform each step:

Step 1: Subtract 7 from both sides:
$5x + 7 - 7 = 32 - 7$

This simplifies to:
$5x = 25$

Step 2: Divide both sides by 5 to isolate $x$:
$x = \frac{25}{5}$

Perform the division:
$x = 5$

Hence, the solution to the equation $5x + 7 = 32$ is $x = 5$.

We will solve the given linear equation $8x + 4 = 4$ step-by-step:

Step 1: Subtract 4 from both sides of the equation to begin isolating the variable $x$:

$8x + 4 - 4 = 4 - 4$

This simplifies to:

$8x = 0$

Step 2: Divide both sides of the equation by 8 to solve for $x$:

$\frac{8x}{8} = \frac{0}{8}$

This results in:

$x = 0$

Therefore, the solution to the equation is $x = 0$.

To solve the equation $14x - 6 = 134$, follow these steps:

• Step 1: Isolate the term involving $x$. We do this by adding 6 to both sides of the equation to eliminate the constant term:

$14x - 6 + 6 = 134 + 6$

This simplifies to:

$14x = 140$

• Step 2: Solve for $x$ by dividing both sides by 14 (the coefficient of $x$):

$\frac{14x}{14} = \frac{140}{14}$

This gives us:

$x = 10$

Therefore, the solution to the problem is $x = 10$.

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

• Step 1: Combine like terms on the left side of the equation.
• Step 2: Isolate the variable $x$ on one side of the equation.
• Step 3: Solve for $x$ and simplify the result.

Now, let's work through each step:
Step 1: The left side of the equation is $800 - 2x - x$. Combine the terms with $x$:
This becomes $800 - 3x = 803$.

Step 2: Subtract 800 from both sides to isolate the term with $x$:
$800 - 3x - 800 = 803 - 800$
This simplifies to $-3x = 3$.

Step 3: Divide both sides by -3 to solve for $x$:
$x = \frac{3}{-3}$
Thus, $x = -1$.

Therefore, the solution to the problem is $x = -1$.

We will solve the equation step by step:

Given equation:
$-3x + 8 = 7x - 12$

• Step 1: Move all $x$-terms to one side by adding $3x$ to both sides.
  $-3x + 3x + 8 = 7x + 3x - 12$
  This simplifies to:
  $8 = 10x - 12$
• Step 2: Move constant terms to the opposite side by adding $12$ to both sides.
  $8 + 12 = 10x - 12 + 12$
  Which simplifies to:
  $20 = 10x$
• Step 3: Solve for $x$ by dividing both sides by $10$.
  $\frac{20}{10} = \frac{10x}{10}$
  This gives us:
  $x = 2$

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

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 equation $\frac{2}{5}x = \frac{3}{8}$, we need to isolate $x$. We can achieve this by multiplying both sides by the reciprocal of $\frac{2}{5}$.

Step 1: Multiply both sides by $\frac{5}{2}$, which is the reciprocal of $\frac{2}{5}$:

$\frac{5}{2} \times \frac{2}{5}x = \frac{5}{2} \times \frac{3}{8}$

Step 2: Simplify the left side. The $\frac{5}{2}$ and $\frac{2}{5}$ cancel each other out:

$x = \frac{5 \times 3}{2 \times 8}$

Step 3: Simplify the right side by multiplying the numerators and denominators:

$x = \frac{15}{16}$

Therefore, the solution to the equation is $\boxed{\frac{15}{16}}$, which matches choice 3.

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 this problem, we need to isolate $x$ by performing the following steps:

• Step 1: Start with the given equation $10x = \frac{6}{11}$.
• Step 2: Divide both sides of the equation by 10 to solve for $x$. $x = \frac{\frac{6}{11}}{10}$
• Step 3: Simplify the fraction on the right. Dividing a fraction by a whole number involves multiplying the denominator by that number: $x = \frac{6}{11 \times 10} = \frac{6}{110}$
• Step 4: Reduce the fraction $\frac{6}{110}$. The greatest common divisor of 6 and 110 is 2: $x = \frac{6 \div 2}{110 \div 2} = \frac{3}{55}$

Thus, the solution to the problem is $x = \frac{3}{55}$.