Solving Equations by using Addition/ Subtraction: Solving an equation by multiplying/dividing both sides

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 $10 + 9x = 91$, 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 $5x + 6 = 56$, 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 $4a + 5 - 24 + a = -2a$, 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 problem, we will use the following steps:

• Step 1: Simplify the equation by combining like terms.
• Step 2: Isolate the variable $m$ using algebraic methods.
• Step 3: Solve for $m$ and verify the solution.

Let's begin:

Step 1: Simplify the equation $m + 3m - 17m + 6 = -20$.
Combine the coefficients of $m$:

$(1 + 3 - 17)m + 6 = -20$

This simplifies to:

$-13m + 6 = -20$

Step 2: Isolate $m$.
Subtract 6 from both sides:

$-13m + 6 - 6 = -20 - 6$

Simplifies to:

$-13m = -26$

Step 3: Solve for $m$ by dividing both sides by -13:

$m = \frac{-26}{-13}$

The division simplifies to:

$m = 2$

Therefore, the solution to the problem is $m = 2$, which corresponds to choice 2 in the given options.

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 this problem, we will simplify and solve the linear equation step-by-step:

1. Start with the given equation:
$2x + 4 + 28 - 3x = x$

2. Combine like terms on the left side:
$(2x - 3x) + 4 + 28 = x$

3. This simplifies to:
$-x + 32 = x$

4. Move all terms involving $x$ to one side of the equation by adding $x$ to both sides:
$32 = 2x$

5. Finally, divide both sides by 2 to solve for $x$:
$x = \frac{32}{2}$

6. Simplify to get the solution:
$x = 16$

Therefore, the solution to the problem is $\mathbf{x = 16}$.

To solve the equation $3x + 4 + 8x - 15 = 0$, we begin by combining the terms that involve $x$ and the constant terms:

Step 1: Combine like terms.
The terms involving $x$ are $3x$ and $8x$. Adding these yields:

The constant terms are $+4$ and $-15$. Combining these gives:

Thus, the equation becomes:

Step 2: Solve for $x$.
To isolate $x$, add 11 to both sides of the equation:

Now, divide both sides by 11:

Therefore, the solution to the equation is $x = 1$.

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 $x = \frac{1}{2}$.

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 $x = 2$.

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:

$-6=x$

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

• Simplify the term $2y \cdot \frac{1}{y}$
• Rearrange the equation to group similar terms
• Solve for $y$

Now, let's work through each step:

Step 1: Simplify the expression $2y \cdot \frac{1}{y}$.

The term $2y \cdot \frac{1}{y}$ simplifies directly to $2$ since $y$ in the numerator and denominator cancel each other out assuming $y \neq 0$. Therefore, the equation becomes:

$2 - y + 4 = 8y$

Step 2: Combine like terms on the left-hand side:

$2 + 4 = 6$, so the equation now is $6 - y = 8y$.

Step 3: Rearrange the equation to isolate $y$ on one side. Add $y$ to both sides to get rid of the negative $y$:

$6 = 8y + y$

This simplifies to:

$6 = 9y$

Step 4: Solve for $y$ by dividing both sides by 9:

$y = \frac{6}{9}$

Simplify the fraction to get:

$y = \frac{2}{3}$

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