Solving Equations by using Addition/ Subtraction: Combining like terms

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 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 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 $3x + 4 + 8x - 15 = 0$, 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 $x = 1$.

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 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 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 $\frac{2}{3}$.