Simplifying and Combining Like Terms: Equations with variables on both sides

Let's solve the equation $-16 + a = -17$ by isolating the variable $a$.

To isolate $a$, add 16 to both sides of the equation to cancel out the $-16$:

$-16 + a + 16 = -17 + 16$

This simplification results in:

$a = -1$

Thus, the solution to the equation $-16 + a = -17$ is $a = -1$.

If we review the answer choices given, the correct answer is Choice 4, $-1$.

The solution to the problem is $a = -1$.

To solve the equation $x + x = 8$, follow these steps:

• Step 1: Combine like terms. Since the left side of the equation is $x + x$, it can be simplified to $2x$. This gives us the equation $2x = 8$.
• Step 2: Solve for $x$ by isolating it. Divide both sides of the equation by 2 to get $x$.
• Performing the division gives $x = \frac{8}{2}$.
• Step 3: Calculate the result of the division. $\frac{8}{2} = 4$.

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

First we will move terms so that -b remains remains on the left side of the equation.

We'll move 8 to the right-hand side, making sure to retain the plus and minus signs accordingly:

$-b=6-8$

Then we will subtract as follows:

$-b=-2$

Finally, we will divide both sides by -1 (be careful with the plus and minus signs when dividing by a negative):

$\frac{-b}{-1}=\frac{-2}{-1}$

$b=2$

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

• Step 1: Combine like terms.
• Step 2: Simplify and isolate the variable.
• Step 3: Solve for the variable.

Let's address each step in detail:
Step 1: Combine the like terms on the left side of the equation.
The original equation is: $2 + 4y - 2y = 4$ Combine the terms involving $y$:
$4y - 2y = 2y$ The equation now becomes:
$2 + 2y = 4$ Step 2: Simplify the equation to isolate $2y$.
Subtract 2 from both sides to begin the process of isolating $y$:
$2y = 4 - 2$ Simplify the right side:
$2y = 2$ Step 3: Solve for $y$ by dividing both sides by 2:
$y = \frac{2}{2}$ This simplifies to:
$y = 1$ Thus, the solution to the equation is: $y = 1$.

We will rearrange the equation so that x remains on the left side and we will move similar elements to the right side.

Remember that when we move a positive number, it will become a negative number, so we will get:

$x=3-5$

$x=-2$

To solve the equation $-3 + x = -8$, we need to isolate the variable $x$. We can do this by eliminating the constant term on the side with the variable.

Step 1: Add 3 to both sides of the equation to cancel out the $-3$ next to $x$.
This gives us:

$-3 + x + 3 = -8 + 3$

Step 2: Simplifying both sides of the equation results in:

$x = -5$

Therefore, the solution to the equation is $x = -5$, which matches choice 2.

To solve the equation $y - 4\frac{2}{3} = 8$, follow these steps:

• Step 1: Identify the given equation:
  $y - 4\frac{2}{3} = 8$.
• Step 2: Solve for $y$ by eliminating the subtraction of $4\frac{2}{3}$. We do this by adding $4\frac{2}{3}$ to both sides of the equation:
  $y - 4\frac{2}{3} + 4\frac{2}{3} = 8 + 4\frac{2}{3}$.
• Step 3: Simplify the left side:
  Since $4\frac{2}{3} - 4\frac{2}{3} = 0$, the left side simplifies to just $y$.
  Thus, $y = 8 + 4\frac{2}{3}$.
• Step 4: Simplify the right side by converting the mixed number to an improper fraction:
  Convert $4\frac{2}{3}$ to an improper fraction: Multiply 4 by 3 and add 2, giving $\frac{14}{3}$.
  Therefore, add $8 + \frac{14}{3}$. Convert 8 to a fraction with denominator 3, which is $\frac{24}{3}$.
  Now, add these fractions:
  $\frac{24}{3} + \frac{14}{3} = \frac{38}{3}$.
• Step 5: Convert back to a mixed number (if desired):
  $\frac{38}{3} = 12\frac{2}{3}$.

Therefore, the solution to the problem is $y = 12\frac{2}{3}$.

The equation we need to solve is $6x + 18 + 2x = 6 - 4$.

Step 1: Simplify each side of the equation.
On the left side, we have two like terms involving $x$: $6x$ and $2x$. We can combine these terms:

$6x + 2x = 8x$.

Thus, the equation becomes:

$8x + 18 = 6 - 4$.

On the right side, simplify $6 - 4$ to get:

$2$.

The equation now reads:

$8x + 18 = 2$.

Step 2: Isolate the variable $x$.
Subtract 18 from both sides to move the constant term on the right side:

$8x + 18 - 18 = 2 - 18$.

This simplifies to:

$8x = -16$.

Next, divide both sides by 8 to solve for $x$:

$x = \frac{-16}{8}$.

This simplifies to:

$x = -2$.

Therefore, the solution to the equation $6x + 18 + 2x = 6 - 4$ is $x = -2$.

To solve the equation $y + 10 - 3y = -150$, follow these steps:

Step 1: Combine like terms on the left side of the equation:

$y + 10 - 3y$ simplifies to $-2y + 10$.

Now the equation is:

$-2y + 10 = -150$.

Step 2: Subtract 10 from both sides to begin isolating $-2y$:

$-2y + 10 - 10 = -150 - 10$.

This simplifies to:

$-2y = -160$.

Step 3: Divide both sides by $-2$ to solve for $y$:

$\frac{-2y}{-2} = \frac{-160}{-2}$.

Thus, $y = 80$.

Therefore, the solution to the problem is $y = 80$.

To solve the equation $2b + 16 + b = -2$, we'll follow these steps:

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

Now, let's work through each step:

Step 1: Combine the terms with the variable $b$ on the left side. We have $2b + b = 3b$. Thus, the equation becomes:

$3b + 16 = -2$

Step 2: Subtract 16 from both sides to isolate the term with the variable:

$3b + 16 - 16 = -2 - 16$

Which simplifies to:

$3b = -18$

Step 3: Divide both sides by 3 to solve for $b$:

$b = \frac{-18}{3}$

The solution is:

$b = -6$

Therefore, the solution to the equation is $b = -6$, corresponding to choice 1.

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

• Step 1: Simplify the equation by combining like terms.
• Step 2: Isolate the variable $a$ to find its value.
• Step 3: Verify the solution by substitution.

Now, let's work through each step:
Step 1: The given equation is $19 = 3a - 6 + 4a - 2a$.
First, combine the like terms involving $a$:
$3a + 4a - 2a = 5a$.
Now, the equation simplifies to $19 = 5a - 6$.

Step 2: Isolate $a$ by first adding 6 to both sides of the equation to eliminate the constant term on the right side:
$19 + 6 = 5a$
$25 = 5a$.

Next, divide both sides by 5 to solve for $a$:
$a = \frac{25}{5}$
$a = 5$.

Step 3: Verify the solution:
Substitute $a = 5$ back into the original equation:
$19 = 3(5) - 6 + 4(5) - 2(5)$
Calculate each term: $3(5) = 15$, $4(5) = 20$, and $2(5) = 10$.
Therefore, $19 = 15 - 6 + 20 - 10$, which simplifies to $19 = 19$, confirming our solution is correct.

Therefore, the solution to the problem is $a = 5$.

To solve the equation $3x - 18 + 2x = 32$, we begin by simplifying the left side:

• Step 1: Combine like terms for the variable $x$:
  The terms involving $x$ are $3x$ and $2x$. Combining them gives $5x$. So the equation becomes $5x - 18 = 32$.
• Step 2: Isolate the term with the variable:
  Add 18 to both sides of the equation to move the constant term to the right side:
  $5x - 18 + 18 = 32 + 18$
  This simplifies to $5x = 50$.
• Step 3: Solve for $x$:
  To isolate $x$, divide both sides by 5:
  $\frac{5x}{5} = \frac{50}{5}$
  This simplifies to $x = 10$.

Therefore, the solution to the equation is $\mathbf{x = 10}$.

To solve the equation $3 = 5 - x$, we need to isolate $x$. We'll perform a sequence of simple algebraic manipulations:

• Step 1: Add $x$ to both sides of the equation to move $x$ to the left side. This gives us:

$3 + x = 5$

• Step 2: Subtract 3 from both sides of the equation to solve for $x$:

$x = 5 - 3$

• Step 3: Simplify the right side:

$x = 2$

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

To solve the equation $-3 - x = 4$, let's follow these steps:

• Start by isolating $x$. We will first eliminate the $-x$ term from the left side by adding $x$ to both sides of the equation. This gives us:

$-3 = 4 + x$

• Now, subtract $4$ from both sides to solve for $x$. This step ensures $x$ is isolated:

$-3 - 4 = x$

• Calculate the result on the left side:

$-7 = x$

Hence, the solution to the given equation is $x = -7$.

Reviewing the choices, the correct choice is , which is $-7$.

First, we divide both sections by 8:

$\frac{8(-2-x)}{8}=\frac{16}{8}$

Keep in mind that the 8 in the fraction of the left section is reduced, so the equation we get is:

$-2-x=2$

We move the minus 2 to the right section and maintain the plus and minus signs accordingly:

$-x=2+2$

$-x=4$

We divide both sides by minus 1 and maintain the plus and minus signs accordingly when we divide:

$\frac{-x}{-1}=\frac{4}{-1}$

$x=-4$

To solve this linear equation, follow these steps:

• Step 1: Move all terms involving $x$ to one side of the equation.
• Step 2: Move constant terms to the opposite side.
• Step 3: Simplify the equation to isolate $x$.
• Step 4: Solve for $x$ by dividing by its coefficient.

Let's work through it:

Start with the given equation:
$5 - 3x = 8x - 17$

Step 1: Add $3x$ to both sides to move all $x$ terms to the right side:
$5 = 11x - 17$

Step 2: Add $17$ to both sides to move the constants to the left side:
$22 = 11x$

Step 3 & 4: Divide both sides by $11$ to isolate $x$:
$x = \frac{22}{11}$

Simplifying the fraction gives:
$x = 2$

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

To solve for $x$ in the equation $74 - 6x + 3 = 8x + 5x - 18$, follow these steps:

• Step 1: Simplify both sides of the equation.

On the left side:

$74 - 6x + 3 = 77 - 6x$ (Combining the constants)

On the right side:

$8x + 5x - 18 = 13x - 18$ (Combining the $x$ terms)

• Step 2: Set the simplified expressions equal.

$77 - 6x = 13x - 18$

• Step 3: Rearrange the equation to isolate terms with $x$.

Adding $6x$ to both sides:

$77 = 13x + 6x - 18$

$77 = 19x - 18$ (Combining the $x$ terms)

• Step 4: Solve for $x$.

Adding 18 to both sides to get rid of the constant on the right:

$77 + 18 = 19x$

$95 = 19x$

Dividing both sides by 19 to solve for $x$:

$x = \frac{95}{19} = 5$

Thus, the solution to the equation is $x = 5$.

To solve the equation $-33x + 45 - 58 = 38x + 144 - 15$, we will simplify both sides:

• First, combine like terms on the left side: $45 - 58 = -13$.
• This gives us: $-33x - 13 = 38x + 144 - 15$.
• Now, simplify the right side: $144 - 15 = 129$.
• The equation now is: $-33x - 13 = 38x + 129$.

Next, we'll move all $x$-terms to one side:

• Add $33x$ to both sides: $-33x + 33x - 13 = 38x + 33x + 129$.
• This simplifies to: $-13 = 71x + 129$.

Now, isolate the $x$-term:

• Subtract 129 from both sides: $-13 - 129 = 71x$.
• This results in: $-142 = 71x$.

Finally, solve for $x$ by dividing both sides by 71:

• $x = -\frac{142}{71}$.
• Simplifying this fraction: $x = -2$.

The correct value of $x$ is $x = -2$. This corresponds to choice 3.

To solve the given linear equation $-31 + 48x + 46 = 83x - 85 + 15x$, we'll follow these steps:

• Step 1: Simplify both sides by combining like terms.
• Step 2: Move all $x$-terms to one side and constant terms to the other.
• Step 3: Solve for $x$.

Now, let's work through each step:

Step 1: Simplify both sides of the equation:
On the left side, combine like terms: $-31 + 46 = 15$. Thus, the left side becomes $15 + 48x$.
On the right side, combine the $x$-terms: $83x + 15x = 98x$. The right side becomes $98x - 85$.

The equation now reads: $15 + 48x = 98x - 85$.

Step 2: Move all $x$-terms to one side and constant terms to the other:
Subtract $48x$ from both sides: $15 = 98x - 48x - 85$.
Simplify the $x$-terms: $98x - 48x = 50x$. Thus, $15 = 50x - 85$.

Add 85 to both sides: $15 + 85 = 50x$, resulting in $100 = 50x$.

Step 3: Solve for $x$ by dividing both sides by 50:
$x = \frac{100}{50} = 2$.

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

To solve this equation, we'll proceed as follows:

• Step 1: Simplify both sides of the equation by combining like terms.
• Step 2: Move all terms with $x$ to one side of the equation.
• Step 3: Isolate the variable $x$ and solve for it.

Now, let's follow these steps in detail:

Step 1: Simplify each side of the equation by combining like terms.

Left side: $36x - 52 + 8x$ simplifies to $(36x + 8x) - 52 = 44x - 52$.

Right side: $19x + 54 - 31$ simplifies to $19x + (54 - 31) = 19x + 23$.

Thus, the equation becomes:

$44x - 52 = 19x + 23$

Step 2: Move all $x$ terms to one side.

Subtract $19x$ from both sides:

$44x - 19x - 52 = 23$

This simplifies to:

$25x - 52 = 23$

Step 3: Isolate the variable $x$.

Add 52 to both sides:

$25x = 23 + 52$

This gives $25x = 75$.

Finally, divide both sides by 25:

$x = \frac{75}{25}$

Thus, $x = 3$.

Therefore, the solution to the problem is $x = 3$, which corresponds to choice 2.