Solving Equations Using All Methods: Equations with variables on both sides

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

In order 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 reach a state where we have only one $X$, not $5X$,
Thus we'll divide both sides of the equation by the coefficient of the unknown (in this case - 5).

$X={8\over5}$

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 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 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 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 $x = 5$.

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 this problem, we'll follow these steps:

• Step 1: Simplify both sides of the given equation.
• Step 2: Isolate the variable $x$ on one side of the equation.
• Step 3: Solve for $x$.

Now, let's work through each step:

Step 1: Simplify both sides of the equation.

The original equation is $54x - 36x + 34 = 39 + 5x - 18$.

On the left side, combine like terms: $54x - 36x = 18x$.

So, the equation becomes $18x + 34 = 39 + 5x - 18$.

Simplify the right side: $39 - 18 = 21$.

This gives us $18x + 34 = 21 + 5x$.

Step 2: Isolate the variable $x$ on one side.

Subtract $5x$ from both sides to get all $x$ terms on one side:

$18x - 5x + 34 = 21$.

This simplifies to $13x + 34 = 21$.

Subtract 34 from both sides to move constant terms to the other side:

$13x = 21 - 34$.

This simplifies to $13x = -13$.

Step 3: Solve for $x$.

Divide both sides by 13 to solve for $x$:

$x = \frac{-13}{13}$.

This simplifies to $x = -1$.

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

To solve this problem, let's break down the equation step-by-step:

Start with the original equation:

$\frac{3}{11} - \frac{8}{12}x + \frac{1}{3}x = \frac{1}{2} + \frac{2}{4} - \frac{22}{24}x$

Step 1: Simplify the fractions where possible.

• $\frac{8}{12} = \frac{2}{3}$: Simplifying $\frac{8}{12}x$ gives us:$-\frac{2}{3}x$
• $\frac{2}{4} = \frac{1}{2}$
• $\frac{22}{24}$ simplifies to $\frac{11}{12}$: So, $-\frac{22}{24}x$ becomes $-\frac{11}{12}x$

The equation now looks like this:

$\frac{3}{11} - \frac{2}{3}x + \frac{1}{3}x = \frac{1}{2} + \frac{1}{2} - \frac{11}{12}x$

Step 2: Combine like terms.

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

So, the equation simplifies to:

$\frac{3}{11} - \frac{1}{3}x = 1 - \frac{11}{12}x$

Step 3: Move all terms involving $x$ to one side:

Add $\frac{1}{3}x$ to both sides:

$\frac{3}{11} = 1 - \frac{11}{12}x + \frac{1}{3}x$

Combine the terms with $x$:

• $-\frac{11}{12}x + \frac{1}{3}x = -\frac{11}{12}x + \frac{4}{12}x = -\frac{7}{12}x$

Thus, the equation is:

$\frac{3}{11} = 1 - \frac{7}{12}x$

Step 4: Isolate $x$:

Subtract 1 from both sides:

$\frac{3}{11} - 1 = -\frac{7}{12}x$

Convert $-1$ to a fraction with a common denominator to the left side:$\frac{3}{11} - \frac{11}{11} = \frac{3 - 11}{11} = -\frac{8}{11}$

Now the equation is:

$-\frac{8}{11} = -\frac{7}{12}x$

Multiply both sides by the reciprocal of $-\frac{7}{12}$ to solve for $x$:

$x = -\frac{8}{11} \cdot -\frac{12}{7} = \frac{8 \cdot 12}{11 \cdot 7} = \frac{96}{77}$

Thus, the solution to the equation is $\boxed{\frac{96}{77}}$.

To solve the equation $0.3x - 4.5 + 7.4x = 3.8x - 3.5 + 1.4$, we will follow these steps:

• Step 1: Simplify both sides of the equation by combining like terms.
• Step 2: Isolate all terms involving $x$ on one side and constants on the other side of the equation.
• Step 3: Solve for $x$ by dividing both sides by the coefficient of $x$.

Let's work through each step:

Step 1: Simplify both sides of the equation.
On the left side, combine like terms: $0.3x + 7.4x = 7.7x$.
Thus, the equation becomes:

$7.7x - 4.5 = 3.8x - 3.5 + 1.4$

Simplify the right side:

$3.8x - 3.5 + 1.4 = 3.8x - 2.1$

The equation now is:

$7.7x - 4.5 = 3.8x - 2.1$

Step 2: Isolate the $x$-terms on one side.
Subtract $3.8x$ from both sides to get:

$7.7x - 3.8x - 4.5 = -2.1$

Which simplifies to:

$3.9x - 4.5 = -2.1$

Now, add 4.5 to both sides to isolate the $x$-term:

$3.9x = -2.1 + 4.5$

$3.9x = 2.4$

Step 3: Solve for $x$.
Divide both sides by 3.9:

$x = \frac{2.4}{3.9}$

$x = 0.6153846153...$

Rounding to two decimal places, we find:

$x = 0.61$

Therefore, the solution to the problem is $x = 0.61$.

To solve this problem, let's carefully simplify and solve the given equation:

• Step 1: Simplify each side of the equation.
  Left side: $7.21 + 11.5x - 3.4x$ which simplifies to $7.21 + (11.5 - 3.4)x = 7.21 + 8.1x$.
  Right side: $8.11x - 12.4 + 3.8$ which simplifies to $8.11x - 8.6$.
• Step 2: Set up the simplified equation.
  We have $7.21 + 8.1x = 8.11x - 8.6$.
• Step 3: Move terms involving $x$ to one side and constants to the other.
  Subtract $8.1x$ from both sides: $7.21 = 8.11x - 8.6 - 8.1x$.
  This simplifies to $7.21 = 0.01x - 8.6$.
• Step 4: Solve for $x$.
  Add $8.6$ to both sides to isolate terms with $x$: $7.21 + 8.6 = 0.01x$, which simplifies to $15.81 = 0.01x$.
  Divide both sides by 0.01 to solve for $x$:
  $x = \frac{15.81}{0.01} = 1581$.

Therefore, the solution to the problem is $x = 1581$.

To solve this linear equation, follow these steps:

• Simplify both sides of the equation by combining like terms.
• Move all terms involving $x$ to one side of the equation.
• Isolate $x$ and solve for $x$.

Let's solve the equation step by step:

Step 1: Simplify both sides of the equation:
Combine the terms involving $x$:
$7.23 - 14x + 15.1x = 3.1x - 8.4$ simplifies to
$7.23 + (15.1x - 14x) = 3.1x - 8.4$.
This results in:
$7.23 + 1.1x = 3.1x - 8.4$.

Step 2: Move all terms involving $x$ to one side:
Subtract $1.1x$ from both sides to bring all $x$-terms to one side:
$7.23 = 3.1x - 1.1x - 8.4$.
Simplifies to:
$7.23 = 2x - 8.4$.

Step 3: Solve for $x$:
Add $8.4$ to both sides to isolate the constant term:
$7.23 + 8.4 = 2x$.
This gives us:
$15.63 = 2x$.
Now, divide both sides by 2 to solve for $x$:
$x = \frac{15.63}{2} = 7.815$.

Therefore, the solution to the equation is $x = 7.815$.

To solve the linear equation $8.51x + 3.4 - 6.14x = 7.51 + 3.8x - 6.1$, we will proceed with these steps:

• Step 1: Combine like terms on the left side.
• Step 2: Combine like terms on the right side.
• Step 3: Isolate $x$ by collecting all $x$ terms on one side of the equation.
• Step 4: Solve for $x$.

Now, let's work through these steps in detail:

Step 1: Combining like terms on the left side:
The left side of the equation is $8.51x + 3.4 - 6.14x$.
Combine the $x$-terms: $(8.51 - 6.14)x + 3.4 = 2.37x + 3.4$.
The left side simplifies to $2.37x + 3.4$.

Step 2: Combining like terms on the right side:
The right side of the equation is $7.51 + 3.8x - 6.1$.
Combine the constant terms: $7.51 - 6.1 = 1.41$.
The right side simplifies to $3.8x + 1.41$.

Step 3: Isolate $x$:
Start with the equation $2.37x + 3.4 = 3.8x + 1.41$.
Subtract $2.37x$ from both sides to have $3.4 = (3.8 - 2.37)x + 1.41$.
This simplifies to $3.4 = 1.43x + 1.41$.

Subtract 1.41 from both sides to isolate the term with $x$:
$3.4 - 1.41 = 1.43x$, resulting in $1.99 = 1.43x$.

Step 4: Solve for $x$:
Divide both sides by $1.43$:
$x = \frac{1.99}{1.43} \approx 1.39$.

Therefore, the solution to the equation is $x = 1.39$.

To solve this problem, we'll proceed through the following steps:

• Step 1: Simplify each fraction in the equation.
• Step 2: Combine like terms on both sides.
• Step 3: Solve for $x$.

Let's work through these steps:

Step 1: Simplify each fraction:
$\frac{100}{25} = 4$, $\frac{144}{12} = 12$, $\frac{33}{11} = 3$, $\frac{56}{8} = 7$, $\frac{35}{7} = 5$, $\frac{18}{2} = 9$.

Now our equation becomes:

$4 + 12x - 3 = 7x - 5x + 9$

Step 2: Simplify and combine like terms:
On the left side: $4 - 3 + 12x = 1 + 12x$
On the right side: $7x - 5x + 9 = 2x + 9$

The equation now is:

$1 + 12x = 2x + 9$

Step 3: Solve for $x$:
Subtract $2x$ from both sides:
$1 + 10x = 9$
Subtract $1$ from both sides:
$10x = 8$
Divide both sides by $10$:
$x = \frac{8}{10} = \frac{4}{5}$

Therefore, the solution to the problem is $x = \frac{4}{5}$.

To solve the equation $\frac{1}{7} - \frac{3}{5}x + \frac{1}{8}x = \frac{1}{9} + \frac{3}{9} - \frac{2}{10}x$, follow these steps:

• Step 1: Simplify constant terms
  Combine the constant terms on the right side: $\frac{1}{9} + \frac{3}{9} = \frac{4}{9}$.
• Step 2: Handle fractions involving $x$ and simplify
  On the left: Combine $-\frac{3}{5}x$ and $\frac{1}{8}x$ to get a single fraction with common denominator 40: $-\frac{3}{5} \times \frac{8}{8}x + \frac{1}{8} \times \frac{5}{5}x = -\frac{24}{40}x + \frac{5}{40}x = -\frac{19}{40}x$.
• Step 3: Isolate terms involving $x$
  Rewrite the equation: $\frac{1}{7} - \frac{19}{40}x = \frac{4}{9} - \frac{2}{10}x$.
  Bring all $x$-terms to the left, and constant terms to the right: $\frac{1}{7} - \frac{4}{9} = -\frac{2}{10}x + \frac{19}{40}x$.
• Step 4: Simplify each side
  For the constants, find a common denominator 63: $\frac{1}{7} \times \frac{9}{9} - \frac{4}{9} \times \frac{7}{7} = \frac{9}{63} - \frac{28}{63} = \frac{-19}{63}$.
  For $x$-terms, common denominator 40: $-\frac{8}{40}x + \frac{19}{40}x = \frac{11}{40}x$.
• Step 5: Solve for $x$
  Combine: $\frac{-19}{63} = \frac{11}{40}x$.
  Solve: $x = \frac{-19}{63} \times \frac{40}{11} = \frac{-760}{693}$.

Therefore, the solution to the problem is $x = -\frac{760}{693}$.

To solve the given linear equation $\frac{1}{8}x - \frac{3}{4} + \frac{1}{9} = -\frac{2}{8} + \frac{3}{4}x - \frac{1}{2}x$, we need to follow these steps:

First, simplify both sides of the equation:

On the left-hand side, which is $\frac{1}{8}x - \frac{3}{4} + \frac{1}{9}$:

• Simplifying, it remains $\frac{1}{8}x - \frac{3}{4} + \frac{1}{9}$.
• Convert $-\frac{3}{4} + \frac{1}{9}$ to a common denominator. The least common multiple of $4$ and $9$ is $36$.
• $-\frac{3}{4} = -\frac{27}{36}$ and $\frac{1}{9} = \frac{4}{36}$, so $-\frac{27}{36} + \frac{4}{36} = -\frac{23}{36}$.
• The left-hand side is now $\frac{1}{8}x - \frac{23}{36}$.

Now, simplify the right-hand side, which is $-\frac{2}{8} + \frac{3}{4}x - \frac{1}{2}x$:

• $-\frac{2}{8} = -\frac{1}{4}$.
• Simplify $\frac{3}{4}x - \frac{1}{2}x$. The common denominator for $\frac{3}{4}$ and $\frac{1}{2}$ is $4$.
• So $\frac{3}{4}x - \frac{1}{2}x = \frac{3}{4}x - \frac{2}{4}x = \frac{1}{4}x$.
• The right-hand side is now $-\frac{1}{4} + \frac{1}{4}x$.

Combine like terms across the equation:

• We aim to move all terms involving $x$ to one side and constants to the other.
• Subtract $\frac{1}{4}x$ from both sides: $\frac{1}{8}x - \frac{1}{4}x - \frac{23}{36} = -\frac{1}{4}$.
• Bring $-\frac{23}{36}$ to the right side: $\frac{1}{8}x - \frac{1}{4}x = -\frac{1}{4} + \frac{23}{36}$.

Simplify and solve for $x$:

• $\frac{1}{8}x - \frac{1}{4}x = \frac{1}{8}x - \frac{2}{8}x = -\frac{1}{8}x$.
• Add $\frac{23}{36}$ to $-\frac{1}{4}$, by finding a common denominator of $36$.
• $-\frac{1}{4} = -\frac{9}{36}$, so $-\frac{9}{36} + \frac{23}{36} = \frac{14}{36} = \frac{7}{18}$.
• Now we have: $-\frac{1}{8}x = \frac{7}{18}$.
• Multiply both sides by $-8$ to solve for $x$: $(-8) \cdot \left(-\frac{1}{8}x\right) = (-8) \cdot \left(\frac{7}{18}\right)$.
• This simplifies to $x = -\frac{56}{18} = -\frac{28}{9}$.
• $-\frac{28}{9}$ can be rewritten as a mixed number: $-\frac{28}{9} = -3\frac{1}{9}$.

Therefore, the solution is:

$x = -3\frac{1}{9}$.

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

• Step 1: Simplify each fraction in the equation.
• Step 2: Combine like terms involving $x$ on one side of the equation and constants on the other.
• Step 3: Solve the simplified equation for $x$.

Let's work through each step together:

Step 1: Simplify each fraction:

• $\frac{36}{4} = 9$,
• $\frac{35}{7} = 5$,
• $\frac{81}{9} = 9$,
• $\frac{16}{8} = 2$,
• $\frac{45}{5} = 9$,
• $\frac{38}{19} = 2$.

With these simplifications, our equation becomes:

$9x + 5 - 9x = 2 + 9x - 2$.

Step 2: Combine like terms.

• The terms $9x$ and $-9x$ cancel out on the left, so we have:
• $5 = 2 + 9x - 2$.
• Simplify the right side: $5 = 9x$.

Step 3: Solve for $x$:

$5 = 9x$

Divide both sides by $9$:

$x = \frac{5}{9}$.

Therefore, the solution to the equation is $x = \frac{5}{9}$.

Let's solve the given equation $8.15x - 13.14 + 5 = 7.1 - 8.4x$ step by step:

Step 1: Simplify both sides of the equation.
On the left-hand side, combine like terms:

$8.15x - 13.14 + 5 = 8.15x - 8.14$

Step 2: Simplify the right-hand side:

$7.1 - 8.4x$ (no combining needed here).

Now the equation is:

$8.15x - 8.14 = 7.1 - 8.4x$

Step 3: Move all $x$-terms to one side and constant terms to the other side.
Add $8.4x$ to both sides:

$8.15x + 8.4x - 8.14 = 7.1$

$16.55x - 8.14 = 7.1$

Step 4: Move the constant term on the left to the right side by adding 8.14:

$16.55x = 7.1 + 8.14$

$16.55x = 15.24$

Step 5: Solve for $x$ by dividing both sides by 16.55:

$x = \frac{15.24}{16.55}$

$x \approx 0.92$

Therefore, the solution to the equation is $x = 0.92$.