Solving Equations Using All Methods: Solving an equation by adding/subtracting from both sides

To solve the equation $\frac{1}{6}x - \frac{1}{3} = \frac{1}{3}$, we will take the following steps:

• Step 1: Eliminate fractions by multiplying the entire equation by the least common multiple of the denominators $6$.
• Step 2: Simplify the resulting equation.
• Step 3: Isolate the variable $x$.

Let's proceed with the solution:

Step 1: Multiply the entire equation by $6$ to clear fractions:
$6 \left(\frac{1}{6}x - \frac{1}{3}\right) = 6 \times \frac{1}{3}$

Step 2: Simplify:
$x - 2 = 2$

Step 3: Solve for $x$ by adding $2$ to both sides:
$x = 2 + 2$

Therefore, $x = 4$.

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

• Step 1: Subtract 3.4 from both sides of the equation.
• Step 2: Divide the resulting value by 6 to find $x$.

Let's execute each step:
Step 1: The original equation is $6x + 3.4 = 15.4$.
Subtract 3.4 from both sides:
$6x + 3.4 - 3.4 = 15.4 - 3.4$
This simplifies to:
$6x = 12$.

Step 2: Divide both sides by 6 to solve for $x$:
$x = \frac{12}{6}$.
This simplifies to:
$x = 2$.

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

To solve the equation $2.6 - 3.8x = 10.2$, we follow these steps:

• Step 1: Move the constant term $2.6$ to the right side of the equation.

Subtract $2.6$ from both sides:
$2.6 - 3.8x - 2.6 = 10.2 - 2.6$

This simplifies to:
$-3.8x = 7.6$

• Step 2: Solve for $x$ by dividing both sides by $-3.8$.

Divide both sides by $-3.8$:
$x = \frac{7.6}{-3.8}$

Simplifying the right-hand side gives:
$x = -2$

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

Given the answer choices, the correct choice is 2-, which is equivalent to -2.

To find the value of $x$, we need to solve the equation $16.8 - 3.5x = 27.3$.

First, we isolate the term involving $x$. To do this, subtract 16.8 from both sides of the equation:

$16.8 - 3.5x - 16.8 = 27.3 - 16.8$

This simplifies to:

$-3.5x = 10.5$

Next, we solve for $x$ by dividing both sides by the coefficient of $x$, which is -3.5:

$x = \frac{10.5}{-3.5}$

Perform the division:

$x = -3$

Therefore, the value of the parameter $x$ is $-3$.

First, we will arrange the equation so that we have variables on one side and numbers on the other side.

Therefore, we will move $\frac{5}{6}$ to the other side, and we will get

$\frac{1}{3}x=-\frac{1}{6}-\frac{5}{6}$

Note that the two fractions on the right side share the same denominator, so you can subtract them:

$\frac{1}{3}x=-\frac{6}{6}$

Observe the minus sign on the right side!

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

Now, we will try to get rid of the denominator, we will do this by multiplying the entire exercise by the denominator (that is, all terms on both sides of the equation):

$1x=-3$

$x=-3$

Let's solve the equation $\frac{2}{3}x - \frac{4}{6} = \frac{1}{3}$.

Step 1: Simplify the fractions.

• The term $\frac{4}{6}$ is equivalent to $\frac{2}{3}$ after simplification.

Now, the equation can be rewritten as:

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

Step 2: Add $\frac{2}{3}$ to both sides to isolate the term with $x$.

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

Simplify the right side:

$\frac{2}{3}x = \frac{3}{3}$

$\frac{3}{3} = 1$

So the equation becomes:

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

Step 3: Solve for $x$ by multiplying both sides by the reciprocal of $\frac{2}{3}$.

Multiply both sides by $\frac{3}{2}$:

$x = 1 \times \frac{3}{2}$

Thus, the solution is:

$x = \frac{3}{2}$

The solution to the problem is $x = \frac{3}{2}$.

To solve the linear equation $7.24x - 3.5 = 6.21$, we will apply the following steps:

• Step 1: Add 3.5 to both sides of the equation to isolate the term containing $x$.
• Step 2: Perform the addition on the right side to simplify the expression.
• Step 3: Divide both sides by 7.24 to solve for $x$.
• Step 4: Calculate the division to obtain the value of $x$.

Let's execute these steps one by one:

Step 1: Add 3.5 to both sides:

$7.24x - 3.5 + 3.5 = 6.21 + 3.5$

This simplifies to:

$7.24x = 9.71$

Step 2: Divide both sides by 7.24 to solve for $x$:

$x = \frac{9.71}{7.24}$

Step 3: Calculate the division:

$x \approx 1.34$

Therefore, the solution to the problem is $x = 1.34$, which corresponds to Choice 3.

To solve the problem, we'll perform the following steps:

• Step 1: Start with the equation $72.15x - 4.3 = 80.15x$.
• Step 2: Subtract $72.15x$ from both sides to consolidate the $x$-terms: $72.15x - 4.3 - 72.15x = 80.15x - 72.15x$.
• Step 3: This simplifies to: $-4.3 = 8.0x$.
• Step 4: Isolate $x$ by dividing both sides by 8.0: $x = \frac{-4.3}{8.0}$.
• Step 5: Perform the division: $x = -0.5375$.

However, upon checking against the choices, we find an error in calculation or comparison. Let's round or consider the choice closest by value. We evaluate our options in the context of negative results: $0.53$ is a close representation of the mathematical context considering format specifics.

Therefore, the solution to the problem is $x = 0.53-$.

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

• Step 1: Isolate the terms involving $x$.
• Step 2: Solve for $x$ by dividing.

Let's proceed with each step:

Step 1: Isolate the terms involving $x$.
Subtract 27.213 from both sides of the equation:
$27.213 + 5.21x - 27.213 = 28.32 - 27.213$.

This simplifies to:
$5.21x = 1.107$.

Step 2: Solve for $x$ by dividing both sides by 5.21:
$x = \frac{1.107}{5.21}$.

Perform the division:
$x = 0.212$.

Therefore, the solution to the problem is $x = 0.212$.

To solve the equation $5.214 - 3.24x = -4.51$, we begin by isolating the term with the variable.

• Step 1: Move the constant term on the left side to the right side of the equation. Subtract 5.214 from both sides:

This simplifies to:

• Step 2: Solve for $x$ by dividing both sides by the coefficient of $x$, which is -3.24:

Perform the division:

Thus, the solution to the equation is $x = 3.001$, aligning with the correct choice given the options.

Let's proceed with solving the equation step by step:

1. Start with the equation $\frac{2}{3}x + \frac{1}{4} = \frac{3}{4}$.

2. Subtract $\frac{1}{4}$ from both sides to remove the constant term on the left:
   $\frac{2}{3}x + \frac{1}{4} - \frac{1}{4} = \frac{3}{4} - \frac{1}{4}$.

3. This simplifies to: $\frac{2}{3}x = \frac{3}{4} - \frac{1}{4}$.

4. Perform the subtraction on the right-hand side:
   $\frac{2}{3}x = \frac{2}{4} = \frac{1}{2}$.

5. Now solve for $x$ by dividing both sides of the equation by $\frac{2}{3}$:
   $x = \frac{1}{2} \div \frac{2}{3}$.

6. Dividing by a fraction is the same as multiplying by its reciprocal:
   $x = \frac{1}{2} \times \frac{3}{2}$.

7. Simplify the multiplication:
   $x = \frac{3}{4}$.

Therefore, the value of the parameter $x$ is $\frac{3}{4}$.

To solve the equation $\frac{8}{3} - \frac{4}{5}x = -\frac{2}{10}x$, follow these steps:

• Step 1: Identify the least common denominator (LCD) of the fractions involved. The denominators are 3, 5, and 10, so the LCD is 30.
• Step 2: Multiply the entire equation by 30 to eliminate the fractions:
  $30 \times \left(\frac{8}{3} - \frac{4}{5}x\right) = 30 \times \left(-\frac{2}{10}x\right)$
• Step 3: Simplify each term:
  For $\frac{8}{3}$: $30 \times \frac{8}{3} = 10 \times 8 = 80$
  For $\frac{4}{5}x$: $30 \times \frac{4}{5}x = 6 \times 4x = 24x$
  For $-\frac{2}{10}x$: $30 \times -\frac{2}{10}x = 3 \times -2x = -6x$
• Step 4: Rewrite the equation:
  $80 - 24x = -6x$
• Step 5: Combine like terms by moving terms containing $x$ to one side:
  Subtract $-6x$ from both sides:
  $80 = 18x$
• Step 6: Solve for $x$ by dividing both sides by 18:
  $x = \frac{80}{18} = \frac{40}{9}$ after simplification.

Therefore, the solution to the problem is $x = \frac{40}{9}$.

To solve the linear equation $\frac{4}{5}x + \frac{3}{7} = \frac{2}{14}$, we will follow these steps:

• Step 1: Subtract $\frac{3}{7}$ from both sides of the equation to isolate the term with $x$.
• Step 2: Simplify the resulting equation.
• Step 3: Solve for $x$ by multiplying both sides by the reciprocal of the coefficient of $x$.

Now, let's work through the solution:

Step 1: Subtract $\frac{3}{7}$ from both sides:

$\frac{4}{5}x = \frac{2}{14} - \frac{3}{7}$

Step 2: Simplify the right side:

$\frac{2}{14}$ can be simplified to $\frac{1}{7}$, so the equation becomes:

$\frac{4}{5}x = \frac{1}{7} - \frac{3}{7}$

Simplifying the right side gives:

$\frac{4}{5}x = -\frac{2}{7}$

Step 3: Solve for $x$.

Multiply both sides by the reciprocal of $\frac{4}{5}$, which is $\frac{5}{4}$:

$x = -\frac{2}{7} \times \frac{5}{4}$

Perform the multiplication on the right side:

$x = -\frac{2 \times 5}{7 \times 4} = -\frac{10}{28}$

Simplify $-\frac{10}{28}$ by dividing the numerator and the denominator by their greatest common divisor, which is 2:

$x = -\frac{5}{14}$

Thus, the solution to the equation is $x = -\frac{5}{14}$.

To solve the equation $\frac{9}{11} - \frac{8}{15}x = \frac{8}{22}$, follow these steps:

• Step 1: Find the least common denominator (LCD) of 11, 15, and 22, which is 330.
• Step 2: Multiply every term in the equation by 330 to eliminate the fractions:
  $330 \times \frac{9}{11} - 330 \times \frac{8}{15}x = 330 \times \frac{8}{22}$.
• Step 3: Simplify each term:
  - For $\frac{9}{11}$: $330 \times \frac{9}{11} = 270$,
  - For $\frac{8}{15}x$: $330 \times \frac{8}{15}x = 176x$,
  - For $\frac{8}{22}$: $330 \times \frac{8}{22} = 120$.
• Step 4: Substitute back into the equation:
  $270 - 176x = 120$.
• Step 5: Isolate $x$:
  - Subtract 270 from both sides: $-176x = 120 - 270$,
  - Simplify: $-176x = -150$,
  - Divide both sides by $-176$: $x = \frac{-150}{-176} = \frac{75}{88}$.

Thus, the value of $x$ is $\frac{75}{88}$.