Solving Equations Using All Methods: Solving an equation by multiplying/dividing both sides

To solve the equation $5x = 25$, we will isolate $x$ using division:

• Divide both sides of the equation by 5:

After performing the division, we get:

Thus, the solution to the equation is $x = 5$.

To solve the equation $\frac{1}{3}x = 9$, we need to isolate the variable $x$. To accomplish this, we can multiply both sides of the equation by 3, the reciprocal of $\frac{1}{3}$.

Step-by-step solution:

• Step 1: Multiply both sides by 3.
  $\left(3 \times \frac{1}{3}\right)x = 3 \times 9$
• Step 2: Simplify the left side.
  This gives us $1x = 27$, since $\left(3 \times \frac{1}{3}\right) = 1$.
• Step 3: Conclude that $x = 27$.

Therefore, the solution to the equation is $x = 27$. This matches choice number 1 from the provided options.

To solve the equation $10 + 3x = 19$, follow these steps:

• Step 1: Subtract 10 from both sides of the equation to begin isolating $x$:
• $10 + 3x - 10 = 19 - 10$
• This simplifies to $3x = 9$.
• Step 2: Divide both sides by 3 to solve for $x$:
• $\frac{3x}{3} = \frac{9}{3}$
• This reduces to $x = 3$.

Therefore, the solution to the problem is $x = 3$.

To solve the equation $24 - 8x = -2x$, we need to isolate $x$. Follow these steps:

• Step 1: Move all terms involving $x$ to one side of the equation. Add $8x$ to both sides to get:
  $24 = 8x - 2x$
• Step 2: Simplify the equation by combining like terms on the right:
  $24 = 6x$
• Step 3: Solve for $x$ by dividing both sides by 6:
  $x = \frac{24}{6}$
• Step 4: Simplify the result:
  $x = 4$

Therefore, the solution to the problem is $\mathbf{x = 4}$.

To solve the given linear equation $33x - 11x = 66$, we will follow these steps:

• Simplify the equation by combining like terms.
• Isolate the variable $x$ to find its value.

Here's how we approach it:

Step 1: Combine like terms on the left-hand side of the equation.

We have $33x - 11x$. By combining these terms, we calculate:

$33x - 11x = (33 - 11)x = 22x$.

Our equation now simplifies to $22x = 66$.

Step 2: Isolate $x$ by dividing both sides of the equation by 22.

When we divide both sides of the equation by 22, we get:

$x = \frac{66}{22}$.

By performing the division, we find $x = 3$.

Therefore, the value of $x$ that satisfies the equation $33x - 11x = 66$ is $x = 3$.

To solve the equation $-8x + 3 = -29$, we'll follow these steps:

• Step 1: Subtract 3 from both sides of the equation to eliminate the constant on the left side.
• Step 2: Divide both sides by $-8$, the coefficient of $x$, to solve for $x$.

Let's apply these steps:
Step 1: Subtract 3 from both sides:
$-8x + 3 - 3 = -29 - 3$
This simplifies to:
$-8x = -32$

Step 2: Divide both sides by $-8$ to isolate $x$:
$\frac{-8x}{-8} = \frac{-32}{-8}$
This results in:
$x = 4$

Therefore, the solution to the equation is $x = 4$, which corresponds to choice 4.

To solve the equation $3x - 5 = 10$, we follow these steps:

• Add $5$ to both sides of the equation to eliminate the $-5$:
  $3x - 5 + 5 = 10 + 5$
  Simplifies to:
  $3x = 15$
• Next, divide both sides of the equation by $3$ to solve for $x$:
  $\frac{3x}{3} = \frac{15}{3}$
  This results in:
  $x = 5$

Therefore, the solution to the equation is $x = 5$.

To solve for $x$ in the equation $6x = 72$, follow these steps:

Step 1: Identify the equation and the coefficient of $x$.
The given equation is $6x = 72$, where the coefficient of $x$ is 6.

Step 2: Isolate $x$ by dividing both sides of the equation by the coefficient (6).
Perform the division: $x = \frac{72}{6}$.

Step 3: Simplify the result.
Calculating $\frac{72}{6}$, we get $x = 12$.

Therefore, the solution to the equation is $x = 12$.

To solve this problem, we will follow the steps outlined below:

• Step 1: Recognize that $\frac{1}{5}x = 12$ gives us $x$ multiplied by $\frac{1}{5}$.
• Step 2: Multiply both sides of the equation by 5 to eliminate the fraction.
• Step 3: Simplify the resulting equation to solve for $x$.

Let's proceed step-by-step:

Step 1: We have the equation $\frac{1}{5}x = 12$.

Step 2: To isolate $x$, multiply both sides of the equation by 5:

$5 \times \frac{1}{5}x = 5 \times 12$

Step 3: Simplify both sides:

• The left side simplifies to $x$ because $5 \times \frac{1}{5} = 1$, so $x$ is left alone.
• The right side becomes $60$, since $5 \times 12 = 60$.

Therefore, the value of $x$ is $60$.

Therefore, the solution to the problem is $x = 60$.

To solve this linear equation, we need to isolate the variable $x$. Here are the steps to follow:

• Step 1: Simplify the equation by dividing both sides by 10. This gives us:

$\frac{8x \cdot 10}{10} = \frac{80}{10}$

This simplifies to:

$8x = 8$

• Step 2: Now, isolate $x$ by dividing both sides by 8:

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

This simplifies to:

$x = 1$

Therefore, the solution to the equation $8x \cdot 10 = 80$ is

$x = 1$.

We start by moving the sections:

5X-15 = 30
5X = 30+15

5X = 45

Now we divide by 5

X = 9

To solve the exercise, we first rewrite the entire division as a fraction:

$\frac{20}{4x}=5$

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

To solve the given equation $4x:30 = 2$, we will follow these steps:

• Step 1: Recognize that $4x:30$ implies $\dfrac{4x}{30} = 2$.

• Step 2: Eliminate the fraction by multiplying both sides of the equation by 30.

• Step 3: Simplify the equation to solve for $x$.

Now, let's work through each step:

Step 1: The equation is written as $\dfrac{4x}{30} = 2$.

Step 2: Multiply both sides of the equation by 30 to eliminate the fraction:
$30 \times \dfrac{4x}{30} = 2 \times 30$

This simplifies to:
$4x = 60$

Step 3: Solve for $x$ by dividing both sides by 4:
$x = \dfrac{60}{4} = 15$

Therefore, the solution to the problem is $x = 15$.

Checking choices, the correct answer is:

$x = 15$

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

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

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

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

3. Calculate $x$:

$x = 3$

To solve the given equation $7x + 5.5 = 19.5$, we'll follow these steps:

• Step 1: Eliminate the constant term from the left side by subtracting 5.5 from both sides of the equation.
• Step 2: Simplify the equation after subtraction to isolate the term with $x$.
• Step 3: Use division to solve for $x$.

Now, let's work through each step:

Step 1: Subtract 5.5 from both sides.

We have:
$7x + 5.5 - 5.5 = 19.5 - 5.5$

This simplifies to:
$7x = 14$

Step 2: Divide both sides by 7 to solve for $x$.

So, we divide by 7:
$\frac{7x}{7} = \frac{14}{7}$

This simplifies to:
$x = 2$

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

To solve the equation $6x \cdot 2 = 24$, 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 $6 \times 2 = 12$) to isolate $x$:

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

3. Calculate $x$:

$x = 2$

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 this problem, we'll follow the procedure of simplifying and solving for $x$:

• Step 1: Simplify both sides of the equation.
• Step 2: Combine like terms and move them to opposite sides to isolate $x$.
• Step 3: Solve for $x$ by performing necessary arithmetic operations.

Now, let's work through each step:

Step 1: Simplify both sides of the equation.
The given equation is $x + 3 - 8x = 4 + 3 - x$.
Combine like terms on each side:
Left side: $x - 8x + 3 = -7x + 3$
Right side: $4 - x + 3 = 7 - x$
So the equation becomes: $-7x + 3 = 7 - x$.

Step 2: Get all terms involving $x$ on one side of the equation.
Add $x$ to both sides to combine the $x$ terms:
$-7x + x + 3 = 7 - x + x$
Simplifies to: $-6x + 3 = 7$

Step 3: Solve for $x$.
Subtract 3 from both sides to isolate terms involving $x$: $-6x + 3 - 3 = 7 - 3$ $-6x = 4$
Now, divide both sides by $-6$ to solve for $x$: $x = \frac{4}{-6} = -\frac{2}{3}$

Therefore, the solution to the problem is $x = -\frac{2}{3}$.

To solve for $x$, let's follow these steps:

• Step 1: Combine like terms on both sides of the equation.
• Step 2: Isolate the $x$ terms on one side.
• Step 3: Solve for $x$.

Let's begin with the left side of the equation:
$-5x + 20 - 3x$ simplifies to $-8x + 20$.

Next, the right side of the equation:
$40 + 2 - 6x$ simplifies to $42 - 6x$.

The equation now is:
$-8x + 20 = 42 - 6x$.

Step 2: Move all terms containing $x$ to one side and constant terms to the other:

First, add $8x$ to both sides to move the $x$ terms together:
$-8x + 8x + 20 = 42 + 2x$
which simplifies to $20 = 42 + 2x$.

Next, subtract $42$ from both sides to get:
$20 - 42 = 2x$
which simplifies to $-22 = 2x$.

Step 3: Solve for $x$ by dividing both sides by 2:
$x = \frac{-22}{2} = -11$.

Therefore, the solution to the problem is $x = -11$.