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 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: $y = 1$.

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 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 $x = 4$.

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 $\mathbf{x = 10}$.

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:

$x=3-5$

$x=-2$

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 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 $y - 4\frac{2}{3} = 8$, 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 $y = 12\frac{2}{3}$.

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 $-3 - x = 4$, 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 $-7$.

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 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.

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 $3 = 5 - x$, 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 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 $x = 2$.

Let's solve the equation step by step:

Given equation: $-22x + 35 - 4x = 31 - 8 + 10x$.

First, simplify both sides by combining like terms.

On the left side:

• Combine all terms with $x$: $-22x - 4x = -26x$.
• The constant term remains: $+35$.
• So, the left side simplifies to: $-26x + 35$.

On the right side:

• Simplify constants: $31 - 8 = 23$.
• The term with $x$ remains: $+10x$.
• So, the right side simplifies to: $23 + 10x$.

The equation now is: $-26x + 35 = 23 + 10x$.

Next, move all terms involving $x$ to one side and constant terms to the other side:

• Subtract $10x$ from both sides: $-26x - 10x + 35 = 23$.
• Combine like terms: $-36x + 35 = 23$.

Now, isolate the $x$ term:

• Subtract 35 from both sides: $-36x = 23 - 35$.
• Simplify the constants: $-36x = -12$.

Finally, solve for $x$ by dividing both sides by $-36$:

• $x = \frac{-12}{-36}$.
• Which simplifies to: $x = \frac{1}{3}$.

Therefore, the solution to the problem is $x = \frac{1}{3}$.

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.

To solve the equation $-45 + 3x + 99 = 5x + 11x + 2$, we'll proceed as follows:

Step 1: Combine like terms on both sides of the equation.

• The left side becomes: $3x + 99 - 45 = 3x + 54$.
• The right side combines terms with $x$: $5x + 11x = 16x$. Thus, the right side is $16x + 2$.

The equation now looks like this: $3x + 54 = 16x + 2$.

Step 2: Move all terms involving $x$ to one side and constant terms to the other side.

Subtract $3x$ from both sides to begin isolating $x$:

• This gives: $54 = 16x - 3x + 2$, simplifying to $54 = 13x + 2$.

Step 3: Isolate $x$.

• Subtract $2$ from both sides: $54 - 2 = 13x$.
• This simplifies to $52 = 13x$.
• Divide both sides by 13 to solve for $x$: $x = \frac{52}{13}$.

Finally, simplify $\frac{52}{13} = 4$.

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

To solve the equation, we will move similar elements to one side.

On the right side, we place the elements with X, while in the left side we place the elements without X.

Remember that when we move sides, the plus and minus signs change accordingly, so we get:

$-9-3=2x+x$

We calculate both sides:$-12=3x$

Finally, divide both sides by 3:

$-\frac{12}{3}=\frac{3x}{3}$

$-4=x$