Solving Equations by using Addition/ Subtraction: Equations with variables on both sides

To solve for $x$, first, we need to get all terms involving $x$ on one side of the equation and constant terms on the other. Start with the original equation:

$2x + 4 = 3x - 5$

Subtract $2x$ from both sides to isolate the term involving $x$ on one side:

$4 = x - 5$

Next, add 5 to both sides to isolate $x$:

$9 = x$

Thus, the value of $x$ is $9$.

To solve the equation $5x + 2 = 4x + 10$, we can simplify and solve for $x$ by following these steps:

• First, let's get all terms involving $x$ on one side and the constant terms on the other. We do this by subtracting $4x$ from both sides:

  $5x + 2 - 4x = 4x + 10 - 4x$

  This simplifies to:

  $x + 2 = 10$

• Next, we need to isolate $x$ by subtracting 2 from both sides:

  $x + 2 - 2 = 10 - 2$

  Which simplifies to:

  $x = 8$

Thus, the solution for $x$ is $8$.

The given equation is: $6x-3=7x+5$

Our goal is to solve for $x$. To achieve this, we'll first get all the terms containing $x$ on one side of the equation and constants on the other side.

Step 1: Subtract $6x$ from both sides to get all $x$ terms on one side:

• $6x - 3 - 6x = 7x + 5 - 6x$

This simplifies to:

• $-3 = x + 5$

Step 2: Next, subtract $5$ from both sides to isolate $x$:

• $-3 - 5 = x + 5 - 5$

This simplifies to:

• $-8 = x$

Therefore, the solution for $x$ is $-8$.

To solve the equation $3x + 5 = 2x + 20$, we need to find the value of $x$ that satisfies this equation. Here are the detailed steps:

• Step 1: Eliminate the variable from one side.
  We want to get all terms involving $x$ on one side and constant terms on the other side. First, subtract $2x$ from both sides of the equation to eliminate $x$ from the right side.

  $3x + 5 - 2x = 2x + 20 - 2x$

  This simplifies to:

  $x + 5 = 20$

• Step 2: Simplify the equation.
  Now, we need to isolate $x$ by removing the constant term from the left side. Subtract 5 from both sides:

  $x + 5 - 5 = 20 - 5$

  This simplifies to:

  $x = 15$

• Step 3: Verify the solution.
  Substitute $x = 15$ back into the original equation to check if it holds true:

  $3(15) + 5 = 2(15) + 20$

  This results in:

  $45 + 5 = 30 + 20$

  $50 = 50$

  Since both sides of the equation are equal,$x = 15$ is indeed the correct solution.

Therefore, the solution to the equation $3x + 5 = 2x + 20$ is $x = 15$.

We start with the equation:
$4x + 4 = 5x + 2$

Our goal is to solve for $x$. To do this, we aim to collect all terms containing $x$ on one side of the equation and constant terms on the other side. First, subtract $4x$ from both sides of the equation to eliminate the $x$ term on the left side:

$4x + 4 - 4x = 5x + 2 - 4x$

This simplifies the equation to:

$4 = x + 2$

Next, subtract $2$ from both sides to isolate the variable $x$ on the right side:

$4 - 2 = x + 2 - 2$

This gives us:

$2 = x$

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

Start by moving the $7x$ term to the left side by subtracting $7x$ from both sides:
$8x - 7x - 1 = 7x + 5 - 7x$
This simplifies to:
$x - 1 = 5$

Next, add$1$ to both sides to isolate $x$:
$x - 1 + 1 = 5 + 1$
Simplifying this, we get:
$x = 6$.

To solve the equation $9x - 3 = 10x + 1$, we need to get all terms with $x$ on one side and constant terms on the other side. Here's how we do it step-by-step:

• First, subtract $9x$ from both sides of the equation to start getting $x$ terms on one side. This gives us: $-3 = x + 1$

• Next, subtract 1 from both sides to isolate $x$. We get: $-3 - 1 = x$

• Simplifying the left side, we find: $x = -4$

Therefore, the solution is $x = -4$.

To solve the equation $x + 3 = -5 + 2x$, we will proceed with these steps:

• Step 1: Simplify and rearrange the terms.
• Step 2: Isolate the variable $x$.
• Step 3: Solve for $x$.

Let's go through each of these steps.

Step 1: Simplify the equation by moving all terms involving $x$ to one side and constant terms to the other. Subtract $x$ from both sides:
$x + 3 - x = -5 + 2x - x$
This simplifies to:
$3 = -5 + x$

Step 2: Add 5 to both sides to isolate $x$:
$3 + 5 = x$

Step 3: Simplify the result:
$8 = x$

Therefore, the solution to the equation is $x = 8$.

To solve the given problem, we'll proceed as follows:

• Step 1: Simplify both sides of the equation.
• Step 2: Check if $x$ can be isolated or analyze if the equation results in contradictions.

Now, let's work through each step:
Step 1: Simplify the left side: $x + 3 - 4x = (1x - 4x) + 3 = -3x + 3$.
Step 2: Simplify the right side: $5x + 6 - 1 - 8x = (5x - 8x) + (6 - 1) = -3x + 5$.

The simplified equation becomes:

$-3x + 3 = -3x + 5$

To solve for $x$, we attempt to isolate $x$. If we add $3x$ to both sides to eliminate the $-3x$ terms, we get:

$3 = 5$

This results in a contradiction, as 3 is not equal to 5, indicating that there is no value of $x$ that can satisfy this equation.

Therefore, the solution to the problem is no solution as indicated by the contradiction.

To solve the equation $6 - 7x = -5x + 8$, we will follow these steps:

• Step 1: Move all terms involving $x$ to one side of the equation by adding $5x$ to both sides.
• Step 2: Simplify both sides of the equation.
• Step 3: Solve for $x$.

Now, let's work through each step:

Step 1: Add $5x$ to both sides to move the $x$-term to the left side:

$6 - 7x + 5x = 8$

Step 2: Simplify the equation by combining the $x$-terms on the left side:

$6 - 2x = 8$

Step 3: To isolate $-2x$ on the left, subtract $6$ from both sides:

$-2x = 8 - 6$

Simplify the right side:

$-2x = 2$

Finally, divide both sides by $-2$ to solve for $x$:

$x = \frac{2}{-2}$

$x = -1$

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

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 $\frac{1}{2}x + 1 = \frac{1}{2}$, follow these steps:

• Step 1: Subtract 1 from both sides of the equation to isolate the term containing $x$.

  $\frac{1}{2}x + 1 - 1 = \frac{1}{2} - 1$

  Simplifying the right side, we have:

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

• Step 2: Simplify the expression on the right side.

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

  So the equation becomes $\frac{1}{2}x = -\frac{1}{2}$.

• Step 3: Solve for $x$ by multiplying both sides of the equation by 2 to eliminate the fraction.

  $2 \times \frac{1}{2}x = 2 \times -\frac{1}{2}$

  This simplifies to $x = -1$.

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

Let's solve the equation $-\frac{1}{4}x + 3 = \frac{1}{2}$ following these steps:

• Step 1: Subtract 3 from both sides to eliminate the constant on the left.

$-\frac{1}{4}x + 3 - 3 = \frac{1}{2} - 3$

$-\frac{1}{4}x = \frac{1}{2} - 3$

Simplifying the right side gives:

$-\frac{1}{4}x = -\frac{5}{2}$

• Step 2: Multiply both sides by $-4$ to isolate $x$.

$x = -\frac{5}{2} \times (-4)$

$x = \frac{20}{2}$

$x = 10$

Thus, we find the solution to be $x = 10$.

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

• Step 1: Manipulate the equation to consolidate terms involving $x$.
• Step 2: Simplify the equation by fixing fractions.
• Step 3: Solve for $x$ to find the solution.

Now, let's work through each step:
Step 1: Start with the original equation, $-\frac{1}{3}x + 5 = \frac{6}{9}x$. First, simplify $\frac{6}{9}$ to $\frac{2}{3}$, giving us the equivalent equation:
$-\frac{1}{3}x + 5 = \frac{2}{3}x.$

Step 2: Move the $-\frac{1}{3}x$ term to the right side to consolidate terms,
$5 = \frac{2}{3}x + \frac{1}{3}x.$

Step 3: Simplify the terms involving $x$ on the right side. $\frac{2}{3}x + \frac{1}{3}x = \frac{3}{3}x = x.$ Thus, the equation becomes:
$5 = x.$

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

We will solve the equation step by step:

Given equation:
$-3x + 8 = 7x - 12$

• Step 1: Move all $x$-terms to one side by adding $3x$ to both sides.
  $-3x + 3x + 8 = 7x + 3x - 12$
  This simplifies to:
  $8 = 10x - 12$
• Step 2: Move constant terms to the opposite side by adding $12$ to both sides.
  $8 + 12 = 10x - 12 + 12$
  Which simplifies to:
  $20 = 10x$
• Step 3: Solve for $x$ by dividing both sides by $10$.
  $\frac{20}{10} = \frac{10x}{10}$
  This gives us:
  $x = 2$

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

First, we arrange the two sections so that the right side contains the values with the coefficient x and the left side the numbers without the x

Let's remember to maintain the plus and minus signs accordingly when we move terms between the sections.

First, we move a$5x$ to the right section and then the 22 to the left side. We obtain the following equation:

$-8-22=10x-5x$

We subtract both sides accordingly and obtain the following equation:

$-30=5x$

We divide both sections by 5 and obtain:

$-6=x$

To solve this exercise, we first need to identify that we have an equation with an unknown,

To solve such equations, the first step will be to arrange the equation so that on one side we have the numbers and on the other side the unknowns.

$2X+7-5X-12=-8X+3$

First, we'll move all unknowns to one side.
It's important to remember that when moving terms, the sign of the number changes (from negative to positive or vice versa).

$2X+7-5X-12+8X=3$

Now we'll do the same thing with the regular numbers.

$2X-5X+8X=3-7+12$

In the next step, we'll calculate the numbers according to the addition and subtraction signs.

$2X-5X=-3X$
$-3X+8X=5X$

$3-7=-4$$$
$-4+12=8$

$5X=8$

At this stage, we want to get to a state where we have only one $X$, not $5X$,
so we'll divide both sides of the equation by the coefficient of the unknown (in this case - 5).

$X={8\over5}$

To solve the equation $-\frac{1}{4} + x = -\frac{1}{2}x$, follow these steps:

• Step 1: Begin by moving the terms involving $x$ to one side of the equation. We can do this by adding $\frac{1}{2}x$ to both sides. This gives:
  $-\frac{1}{4} + x + \frac{1}{2}x = 0$
• Step 2: Recognize that $x + \frac{1}{2}x$ can be combined into a single term:\br $-\frac{1}{4} + \frac{3}{2}x = 0$
• Step 3: Isolate $\frac{3}{2}x$ by adding $\frac{1}{4}$ to both sides, resulting in:
  $\frac{3}{2}x = \frac{1}{4}$
• Step 4: Solve for $x$ by multiplying both sides by $\frac{2}{3}$, which is the reciprocal of $\frac{3}{2}$:
  $x = \frac{1}{4} \cdot \frac{2}{3}$
• Step 5: Simplify the multiplication on the right side:
  $x = \frac{2}{12} = \frac{1}{6}$

Thus, the solution to the equation is $x = \frac{1}{6}$.

To solve the equation $-\frac{1}{4}x + 8 = 3x - \frac{1}{2}$, follow these steps:

• Step 1: Remove fractions by multiplying every term by the least common multiple of the denominators, which is 4.
  This gives $4 \left(-\frac{1}{4}x\right) + 4 \times 8 = 4 \times 3x - 4 \times \frac{1}{2}$, simplifying to $-x + 32 = 12x - 2$.
• Step 2: Move all terms involving $x$ to one side by adding $x$ to both sides.
  This gives $32 = 13x - 2$.
• Step 3: Isolate $x$ by moving the constant to the other side: Add 2 to both sides to obtain $32 + 2 = 13x$, simplifying to $34 = 13x$.
• Step 4: Solve for $x$ by dividing both sides by 13, leading to $x = \frac{34}{13}$.
• Step 5: Simplify the fraction if possible. Here, $\frac{34}{13}$ is already in its simplest form. Converting to a mixed number, we get $2\frac{8}{13}$.

Therefore, the solution to the equation is $x = 2\frac{8}{13}$.

To solve the equation $-\frac{1}{8}x + 4 = 7x - \frac{4}{5}$, follow these steps:

Step 1: Move the $x$ terms to one side of the equation.
Add $\frac{1}{8}x$ to both sides:

$-\frac{1}{8}x + \frac{1}{8}x + 4 = 7x + \frac{1}{8}x - \frac{4}{5} \\ 4 = 7x + \frac{1}{8}x - \frac{4}{5}$

Step 2: Simplify by combining like terms. Combine $x$-terms on the right:

$4 = \left(7 + \frac{1}{8}\right)x - \frac{4}{5}$

Step 3: Express $7 + \frac{1}{8}$ as a single fraction: $\frac{56}{8} + \frac{1}{8} = \frac{57}{8}$.

Step 4: Substitute back:

$4 = \frac{57}{8}x - \frac{4}{5}$

Step 5: Move the constant term to the other side. Add $\frac{4}{5}$ to both sides:

$4 + \frac{4}{5} = \frac{57}{8}x \\ \frac{20}{5} + \frac{4}{5} = \frac{57}{8}x \\ \frac{24}{5} = \frac{57}{8}x$

Step 6: Solve for $x$ by dividing both sides by $\frac{57}{8}$ (multiply by $\frac{8}{57}$):

$x= \frac{24}{5} \times \frac{8}{57} \\ x = \frac{192}{285}$

Therefore, the solution to the problem is $x = \frac{192}{285}$.