Solving an Equation by Multiplication/ Division: Combining like terms

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:

$b=-1$

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 $2 + 3a + 4 = 0$, 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 $2y + 12 - 5y + 30 = 0$, follow these steps:

• Step 1: Simplify the equation by combining like terms.
  Combine the $y$ terms and the constant terms:
  $2y - 5y + 12 + 30 = 0$
• Step 2: Calculate the combined terms.
  $2y - 5y = -3y$
  $12 + 30 = 42$
  Thus, the equation becomes:
  $-3y + 42 = 0$
• Step 3: Isolate the variable $y$.
  Subtract 42 from both sides to get:
  $-3y = -42$
• Step 4: Solve for $y$ by dividing both sides by $-3$:
  $y = \frac{-42}{-3}$
• Step 5: Simplify the fraction:
  $y = 14$

Therefore, the solution to the equation is $y = 14$.

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 $5x + 10 = 3x + 18$, 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 $7x - 3 = 4x + 9$, 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 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 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 this problem, we'll follow these steps:

• Step 1: Combine like terms on the left side of the equation.
• Step 2: Isolate the variable $x$ on one side of the equation.
• Step 3: Solve for $x$ and simplify the result.

Now, let's work through each step:
Step 1: The left side of the equation is $800 - 2x - x$. Combine the terms with $x$:
This becomes $800 - 3x = 803$.

Step 2: Subtract 800 from both sides to isolate the term with $x$:
$800 - 3x - 800 = 803 - 800$
This simplifies to $-3x = 3$.

Step 3: Divide both sides by -3 to solve for $x$:
$x = \frac{3}{-3}$
Thus, $x = -1$.

Therefore, the solution to the problem is $x = -1$.

To solve this problem, we need to find $x$ in the equation:

$20 + 20x - 3x = 88$

Step 1: Combine like terms on the left-hand side of the equation. The terms involving $x$ are $20x$ and $-3x$.

$20x - 3x = 17x$

Thus, the equation becomes:

$20 + 17x = 88$

Step 2: Isolate the $x$-related terms by moving the constant term to the right-hand side. To do this, subtract 20 from both sides:

$17x = 88 - 20$

$17x = 68$

Step 3: Solve for $x$ by dividing both sides of the equation by 17:

$x = \frac{68}{17}$

$x = 4$

Therefore, the solution to the problem is $x = 4$.

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 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.

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 $2 - 5x + 4 - 1x = 0$, we proceed as follows:

• Step 1: Simplify the left side of the equation.

Combine the constant terms $2$ and $4$:

$2 + 4 = 6$

Combine the terms involving $x$:

$-5x - 1x = -6x$

Thus, the equation becomes:

$6 - 6x = 0$

• Step 2: Isolate the variable $x$.

Move $6$ to the other side of the equation by subtracting $6$ from both sides:

$-6x = -6$

Divide both sides by $-6$ to solve for $x$:

$x = \frac{-6}{-6} = 1$

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

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 $-3x + 8 - 11 = 40x + 5x + 9$, we need to combine and simplify terms:

• Simplify each side separately. Start with the right side: $40x + 5x + 9 = 45x + 9$.
• Now simplify the left side: $-3x + 8 - 11 = -3x - 3$.

The equation is now: $-3x - 3 = 45x + 9$. Next, move all $x$-terms to one side and constants to the other side:

• Add $3x$ to both sides: $-3x - 3 + 3x = 45x + 9 + 3x$, which simplifies to: $-3 = 48x + 9$.

Then, move the constant term $9$ to the left side:

• Subtract $9$ from both sides: $-3 - 9 = 48x + 9 - 9$, which simplifies to: $-12 = 48x$.
• Solve for $x$ by dividing both sides by 48: $x = \frac{-12}{48}$.
• Simplify the fraction: $x = -\frac{1}{4}$.

Therefore, the solution to the problem is $x = -\frac{1}{4}$.

To solve the given linear equation, we will follow these steps:

• Step 1: Combine the terms involving $y$.
• Step 2: Simplify the constants on the right side of the equation.
• Step 3: Isolate $y$ to find its value.

Let’s solve the equation $\frac{1}{4}y + \frac{1}{2}y + 5 - 12 = 0$.

Step 1: Combine the like terms that involve $y$.
The coefficients of $y$ are $\frac{1}{4}$ and $\frac{1}{2}$. To combine them, we need a common denominator, which is 4. Therefore:

$\frac{1}{4}y + \frac{1}{2}y = \frac{1}{4}y + \frac{2}{4}y = \frac{3}{4}y$.

Step 2: Simplify the constants.
The equation now becomes $\frac{3}{4}y + 5 - 12 = 0$.
Combine the constants: $5 - 12 = -7$.

The equation simplifies to $\frac{3}{4}y - 7 = 0$.

Step 3: Isolate $y$.
Add 7 to both sides of the equation:
$\frac{3}{4}y = 7$.

To solve for $y$, multiply both sides by the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$:

$y = 7 \times \frac{4}{3} = \frac{28}{3}$.

Convert the fraction to a mixed number: $\frac{28}{3} = 9 \cdot 3 + 1 = 9 \text{ remainder } 1$. Thus, $\frac{28}{3} = 9\frac{1}{3}$.

Therefore, the value of $y$ is $9\frac{1}{3}$.

To solve the linear equation $6x \cdot 2 - 4 + 2x + 2 = 5$, follow these steps:

• Step 1: Simplify the expression on the left-hand side of the equation.
• Step 2: Combine like terms to reduce the equation.
• Step 3: Isolate the variable $x$ to determine its value.

Let's simplify and solve the given equation:

Step 1: Simplify the expression $6x \cdot 2 - 4 + 2x + 2$.
This becomes $12x - 4 + 2x + 2$.

Step 2: Combine like terms.
Combine the terms involving $x$: $12x + 2x = 14x$.
Combine the constants: $-4 + 2 = -2$.
This results in the equation $14x - 2 = 5$.

Step 3: Isolate $x$.
Add 2 to both sides to eliminate the constant on the left:
$14x - 2 + 2 = 5 + 2$.
This simplifies to $14x = 7$.
Next, divide both sides by 14 to solve for $x$:
$x = \frac{7}{14}$.

Simplify the fraction:$x = \frac{1}{2}$.

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

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

• Step 1: Eliminate multiplication by distributing $2x \cdot 4$.
• Step 2: Combine like terms on the left side of the equation.
• Step 3: Isolate the variable $x$ by moving constants to the opposite side.

Let's work through these steps:

Step 1: The given equation is $2x \cdot 4 - 1 + x + 2 = 19$.
Distribute the multiplication on $2x \cdot 4$ to get $8x$:

$8x - 1 + x + 2 = 19$

Step 2: Combine the like terms ($8x$ and $x$):

$9x - 1 + 2 = 19$

Simplify further by combining constants $-1 + 2$ to get:

$9x + 1 = 19$

Step 3: Isolate $x$ by subtracting 1 from both sides:

$9x = 18$

Finally, divide both sides by 9 to solve for $x$:

$x = \frac{18}{9} = 2$

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