Solving Equations Using All Methods: Combining like terms

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 this equation for $x$, we will follow these steps:

• Simplify both sides of the equation by combining like terms.
• Rearrange the equation to isolate the variable $x$.
• Analyze the resultant equation to determine the solution.

Let's break it down:

Step 1: Simplify the left side:
The left side of the equation is $-7 + 3x - 8x$. Combine the like terms $3x$ and $-8x$:
$-7 + (3x - 8x) = -7 - 5x$

Step 2: Simplify the right side:
The right side of the equation is $9 + 3 - 5x$. Combine the constant terms $9$ and $3$:
$(9 + 3) - 5x = 12 - 5x$

Step 3: Set the simplified equation:
Now the equation is:
$-7 - 5x = 12 - 5x$

Step 4: Analyze the equation:
If we attempt to isolate $x$ by adding $5x$ to both sides, we get:
$-7 = 12$

This statement is false. Since the manipulation leads to a false statement without any variable $x$, the original equation has no solution.

Therefore, the equation cannot be true for any real number value of $x$. Thus, the correct answer is: no solution.

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 $x = -11$.

To solve the given equation $3x + 4 + x + 1 = 9$, we'll proceed step-by-step:

• Step 1: Combine like terms on the left side
  Combine the terms with $x$: $3x + x = 4x$.
  Combine the constant terms: $4 + 1 = 5$.
  The equation becomes $4x + 5 = 9$.
• Step 2: Isolate the variable $x$
  Subtract 5 from both sides to move the constant term to the right side:
  $4x + 5 - 5 = 9 - 5$, which simplifies to $4x = 4$.
• Step 3: Solve for $x$
  Divide both sides by 4 to solve for $x$:
  $\frac{4x}{4} = \frac{4}{4}$, which simplifies to $x = 1$.

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

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 $x = -3.73$.

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.