Solving Equations Using All Methods: Opening parentheses

To solve the equation $7 - 8(x + 4) = -3x$, we'll go through these steps:

• Step 1: Distribute $-8$ across the terms inside the parentheses:

$7 - 8 \cdot x - 8 \cdot 4 = -3x$
This simplifies to $7 - 8x - 32 = -3x$.

• Step 2: Simplify both sides by combining like terms:

$7 - 32 = -25$, so the equation becomes:
$-8x - 25 = -3x$.

• Step 3: Isolate the variable $x$ by moving all terms involving $x$ to one side:

To move $-8x$ to the right side, add $8x$ to both sides:
$-25 = -3x + 8x$
This simplifies to:
$-25 = 5x$.

• Step 4: Solve for $x$ by dividing both sides by 5:

$x = \frac{-25}{5}$, which gives us:
$x = -5$.

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

The correct choice matches this solution as option $\boldsymbol{-5}$.

To solve the equation $3(x-2) = 4$, we follow these detailed steps:

• Step 1: Apply the distributive property to the left side of the equation. This means multiplying 3 by each term within the parentheses.
• Step 2: Expand $3(x-2)$ to get $3x - 6$.
• Step 3: The equation is now $3x - 6 = 4$.
• Step 4: To isolate the term containing $x$, add 6 to both sides of the equation. This yields $3x - 6 + 6 = 4 + 6$, simplifying to $3x = 10$.
• Step 5: Divide both sides of the equation by 3 to solve for $x$. So, $x = \frac{10}{3}$.

Therefore, the solution to the problem is $x = \frac{10}{3}$.

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

• Expand both sides using the distributive property.
• Combine like terms and simplify.
• Isolate the variable $x$.

Let's work through the steps in detail:

Step 1: Apply the distributive property:
- Left side: $6(x+4)$ expands to $6x + 24$.
- Right side: $8(x+5)$ expands to $8x + 40$.
Substituting back, the equation becomes:

$6x + 24 - 4 = 8x + 40$

Step 2: Simplify the equation by combining like terms:
- 24 - 4 simplifies to 20 on the left-hand side.
The equation now is:

$6x + 20 = 8x + 40$

Step 3: Isolate the variable $x$:
- First, eliminate $6x$ from the left side by subtracting $6x$ from both sides:

$20 = 2x + 40$

- Next, eliminate 40 from the right side by subtracting 40 from both sides:

$20 - 40 = 2x$
$-20 = 2x$

Step 4: Solve for $x$ by dividing both sides by 2:

$x = \frac{-20}{2}$
$x = -10$

Therefore, the solution to the problem is $x = -10$.

The goal is to solve the linear equation $7(x-4) = (6-x) \times 3$. Follow these steps to find the solution:

• Step 1: Open the parentheses using the distributive property. This gives us: $7(x - 4) = 7x - 28$ $(6 - x) \times 3 = 18 - 3x$ Thus, the equation becomes: $7x - 28 = 18 - 3x$
• Step 2: Combine like terms by getting all $x$-terms on one side and constant terms on the other. $7x + 3x = 18 + 28$ Simplify to get: $10x = 46$
• Step 3: Solve for $x$ by dividing both sides by 10. $x = \frac{46}{10} = 4.6$

Therefore, the solution to the problem is $x = 4.6$.

To solve the given equation $7(x + 5) - 3(x - 2) = 5$, we follow these steps:

• Step 1: Apply the distributive property.
  Distribute 7 over $(x + 5)$ to obtain $7x + 35$.
  Distribute -3 over $(x - 2)$ to obtain $-3x + 6$.
  The equation becomes $7x + 35 - 3x + 6 = 5$.
  
• Step 2: Combine like terms.
  Combine the $x$ terms $7x$ and $-3x$ to get $4x$.
  Combine the constant terms $35$ and $6$ to get $41$.
  The equation simplifies to $4x + 41 = 5$.
  
• Step 3: Isolate the variable $x$.
  Subtract 41 from both sides: $4x + 41 - 41 = 5 - 41$.
  This simplifies to $4x = -36$.
  Divide both sides by 4 to solve for $x$: $x = \frac{-36}{4}$.
  The solution is $x = -9$.

Therefore, the solution to the equation is $x = -9$.

Let's solve the given equation step by step:

We start with the equation:

$7(x-4) + 3 = 5 - 4(x+5)$

Step 1: Apply the distributive property to eliminate the parentheses.

The left-hand side becomes:

$7(x-4) = 7x - 28$, so the entire expression on the left is $7x - 28 + 3$.

The right-hand side becomes:

$-4(x+5) = -4x - 20$, so the whole right side is $5 - 4x - 20$.

Step 2: Simplify both sides.

Simplify the left-hand side:

$7x - 28 + 3 = 7x - 25$.

Simplify the right-hand side:

$5 - 4x - 20 = -4x - 15$.

Now the equation reads:

$7x - 25 = -4x - 15$.

Step 3: Rearrange the equation to isolate $x$ terms on one side and constants on the other.

Add $4x$ to both sides:

$7x + 4x - 25 = -4x + 4x - 15$.

This simplifies to:

$11x - 25 = -15$.

Step 4: Solve for $x$.

Add 25 to both sides:

$11x = 10$.

Divide both sides by 11 to solve for $x$:

$x = \frac{10}{11}$.

Thus, the solution to the equation is $x = \frac{10}{11}$.

Considering the given choices, the correct answer is choice 3: $\frac{10}{11}$.

Let's solve the linear equation $6 + 3(x + 4) = 7 - 3(x - 2)$ step-by-step.

Step 1: Expand the terms using the distributive property.

On the left side: $3(x + 4) = 3x + 12$

On the right side: $-3(x - 2) = -3x + 6$

Substituting back, the equation becomes:

$6 + 3x + 12 = 7 - 3x + 6$

Step 2: Simplify both sides by combining like terms.

Left side: $6 + 12 + 3x = 18 + 3x$

Right side: $7 + 6 - 3x = 13 - 3x$

The equation now is:

$18 + 3x = 13 - 3x$

Step 3: Bring all terms involving $x$ to one side.

Add $3x$ to both sides:

$18 + 3x + 3x = 13$

$18 + 6x = 13$

Step 4: Isolate the variable $x$.

Subtract 18 from both sides:

$6x = 13 - 18$

$6x = -5$

Divide both sides by 6:

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

Therefore, the solution to the problem is $x = -\frac{5}{6}$.

To solve the equation $5(x-4) - 6(7-x) = 5x$, we will simplify both sides and solve for $x$ step-by-step:

Step 1: Distribute Constants inside Parentheses
Apply the distributive property to simplify each group:
- $5(x-4)$ becomes $5x - 20$.
- $-6(7-x)$ becomes $-42 + 6x$.

Resulting Equation:
$5x - 20 - 42 + 6x = 5x$.

Step 2: Combine Like Terms
Combine $5x$ and $6x$, and $-20$ and $-42$:
- $5x + 6x = 11x$.
- $-20 - 42 = -62$.
The equation is now: $11x - 62 = 5x$.

Step 3: Isolate the Variable $x$
Subtract $5x$ from both sides to get the $x$-terms on one side:
$11x - 5x = 62$.
This simplifies to $6x - 62 = 0$.

Step 4: Solve for $x$
Add 62 to both sides to isolate the term with $x$:
$6x = 62$.
Now, divide by 6 to solve for $x$:
$x = \frac{62}{6} = \frac{31}{3} \approx 10.33$.

Therefore, $x \approx 10.33$.

Let's solve the equation $3-4(x-2)=6x+4(3-x)$.

First, apply the distributive property on both sides:

• For the left side: $3 - 4(x - 2)$ becomes $3 - 4x + 8$, which simplifies to $11 - 4x$.
• For the right side: $6x + 4(3 - x)$ becomes $6x + 12 - 4x$, which simplifies to $2x + 12$.

Now the equation is:

$11 - 4x = 2x + 12$

Combine like terms to isolate $x$. First, move all terms containing $x$ to one side and constant terms to the other side:

• Add $4x$ to both sides: $11 = 6x + 12$.
• Subtract 12 from both sides: $11 - 12 = 6x$, which simplifies to $-1 = 6x$.

Finally, solve for $x$ by dividing both sides by 6:

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

Therefore, the solution to the problem is $x = -\frac{1}{6}$, which corresponds to choice 4.

To solve the given equation $3(x+2) = 5(2-x)$, we will follow these steps:

• Step 1: Distribute on both sides of the equation.

For the left side, distribute $3$ over $(x+2)$:

$3(x+2) = 3x + 6$

For the right side, distribute $5$ over $(2-x)$:

$5(2-x) = 10 - 5x$

• Step 2: Rewrite the equation with the expanded terms.

The equation becomes:

$3x + 6 = 10 - 5x$

• Step 3: Combine like terms to isolate $x$.

First, add $5x$ to both sides to get all $x$ terms on the left side:

$3x + 5x + 6 = 10$

This simplifies to:

$8x + 6 = 10$

Next, subtract 6 from both sides to isolate the term with $x$:

$8x = 4$

• Step 4: Solve for $x$.

Divide both sides by 8 to solve for $x$:

$x = \frac{4}{8}$

Simplify the fraction:

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

Therefore, the solution to the equation is $x = \frac{1}{2}$.

We begin by solving the two exercises inside of the parentheses:

$4a\times7=112$

We then divide each of the sections by 4:

$\frac{4a\times7}{4}=\frac{112}{4}$

In the fraction on the left side we simplify by 4 and in the fraction on the right side we divide by 4:

$a\times7=28$

Remember that:

$a\times7=a7$

Lastly we divide both sections by 7:

$\frac{a7}{7}=\frac{28}{7}$

$a=4$

We begin by solving the addition exercise in the right parenthesis:

$(7x+3)+14=238$

We then multiply each of the terms inside of the parentheses by 14:

$(14\times7x)+(14\times3)=238$

Following this we solve each of the exercises inside of the parentheses:

$98x+42=238$

We move the sections whilst retaining the appropriate sign:

$98x=238-42$

$98x=196$

Finally we divide the two parts by 98:

$\frac{98}{98}x=\frac{196}{98}$

$x=2$