Solving an Equation by Multiplication/ Division: Rearranging Equations

First, we arrange the two sections so that the right side contains the values with the coefficient x and the left side the numbers without the x

Let's remember to maintain the plus and minus signs accordingly when we move terms between the sections.

First, we move a$5x$ to the right section and then the 22 to the left side. We obtain the following equation:

$-8-22=10x-5x$

We subtract both sides accordingly and obtain the following equation:

$-30=5x$

We divide both sections by 5 and obtain:

$-6=x$

Let's first arrange the equation so that on the left-hand side we have the terms with the coefficient $b$ and on the right-hand side the numbers without the coefficient $b$.

Remember that when we move terms across the equals sign, the plus and minus signs will change accordingly:

$2b-3b=5-4$

Let's now solve the subtraction exercise on both sides:

$-1b=1$

Finally, we can divide both sides by -1 to find our answer:

$b=-1$

Let's solve the equation $x + 5 = 11x$ step-by-step.

• Step 1: Isolate the variable $x$
  Start by getting all terms involving $x$ on one side of the equation. We can do this by subtracting $x$ from both sides:
  $x + 5 - x = 11x - x$
• This simplifies to:
  $5 = 10x$
• Step 2: Solve for $x$
  Now, divide both sides of the equation by 10 to solve for $x$:
  $\frac{5}{10} = \frac{10x}{10}$
• This further simplifies to:
  $x = \frac{1}{2}$

Therefore, the solution to the equation $x + 5 = 11x$ is $x = \frac{1}{2}$.

Step-by-step solution:

1. Begin with the equation: $5x - 3 = 2$

2. Add 3 to both sides: $5x - 3 + 3 = 2 + 3$, which simplifies to $5x = 5$

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

4. Simplify the division: $x = 1$

Step-by-step solution:

1. Start with the equation: $3x + 6 = 12$

2. Subtract 6 from both sides: $3x + 6 - 6 = 12 - 6$, which simplifies to $3x = 6$

3. Divide both sides by 3 to solve for x: $x = \frac{6}{3}$

4. Simplify the division: $x = 2$

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 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 $10 + 9x = 91$, we'll follow these steps:

• Step 1: Eliminate the constant term on the left side by subtracting 10 from both sides of the equation.

$10 + 9x - 10 = 91 - 10$ $9x = 81$

• Step 2: Solve for $x$ by dividing each side of the equation by the coefficient of $x$, which is 9.

$\frac{9x}{9} = \frac{81}{9}$ $x = 9$

Hence, the value of $x$ is $9$.

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 the equation $2 + 3a + 4 = 0$, follow these steps:

• Step 1: Combine the constant terms on the left side.
  The terms $2$ and $4$ can be combined to get $6$.
  Hence, the equation becomes $3a + 6 = 0$.
• Step 2: Isolate the term with the variable $a$.
  Subtract $6$ from both sides to get $3a = -6$.
• Step 3: Solve for $a$ by dividing both sides by the coefficient of $a$, which is $3$.
  Thus, $a = \frac{-6}{3} = -2$.

Therefore, the solution to the problem is $a = -2$.

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 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 the equation $4a + 5 - 24 + a = -2a$, follow these steps:

• Step 1: Start by combining like terms on the left side of the equation:

$4a + a + 5 - 24 = -2a$

This simplifies to:

$5a - 19 = -2a$

• Step 2: Move all terms involving $a$ to one side of the equation and constant terms to the other side:

Add $2a$ to both sides to collect all terms with $a$:

$5a + 2a = 19$

This simplifies to:

$7a = 19$

• Step 3: Solve for $a$ by dividing both sides by 7:

$a = \frac{19}{7}$

Thus, the value of $a$ is $\frac{19}{7}$, which can be written as a mixed number:

$a = 2\frac{5}{7}$.

Upon verifying with the given choices, the correct answer is choice 1: $2\frac{5}{7}$.

To solve the equation $4x - 7 = 13$, follow these steps:

1. Add 7 to both sides: $4x = 13 + 7$

2. Simplify the right side: $4x = 20$

3. Divide both sides by 4: $x = \frac{20}{4}$

4. Solve: $x = 5$

To solve the equation $5x + 10 = 3x + 18$, follow these steps:

1. Subtract $3x$ from both sides to get:

$5x - 3x + 10 = 18$

2. Simplify the equation:

$2x + 10 = 18$

3. Subtract $10$ from both sides:

$2x = 8$

4. Divide both sides by $2$:

$x = 4$

To solve the equation $5x + 4 = 7x$, we will proceed as follows:

• Step 1: Subtract $5x$ from both sides to simplify the equation.

The equation is:
$5x + 4 - 5x = 7x - 5x$

This simplifies to:
$4 = 2x$

• Step 2: To solve for $x$, divide both sides by 2 to isolate $x$.

Perform the division:
$\frac{4}{2} = \frac{2x}{2}$

This gives us:
$2 = x$

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

To solve the equation $5x + 6 = 56$, we will follow these steps:

• Step 1: Subtract 6 from both sides of the equation to eliminate the constant term on the left-hand side.
• Step 2: Simplify the resulting equation.
• Step 3: Divide both sides by 5 to isolate $x$.

Now, perform each step:

Step 1: Subtract 6 from both sides:
$5x + 6 - 6 = 56 - 6$

Step 2: Simplify both sides:
$5x = 50$

Step 3: Divide both sides by 5 to solve for $x$:
$x = \frac{50}{5}$

Step 4: Simplify the division:

$x = 10$

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

To solve the equation $7x - 3 = 4x + 9$, follow these steps:

1. Subtract $4x$ from both sides to get:

$7x - 4x - 3 = 9$

2. Simplify the equation:

$3x - 3 = 9$

3. Add $3$ to both sides:

$3x = 12$

4. Divide both sides by $3$:

$x=4$

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 $\frac{1}{5}x - 4 = 6$, 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 $x = 50$.