Solving Equations Using All Methods - Examples, Exercises and Solutions

To solve this equation, follow these steps:

• Step 1: Apply the distributive property to the equation:

  $2(4-x) = 2 \times 4 - 2 \times x = 8 - 2x$

• Step 2: Simplify the equation:

  The equation now becomes: $8 - 2x = 8$

• Step 3: Isolate the variable $x$ by simplifying the equation:

  First, subtract 8 from both sides:
  $8 - 2x - 8 = 8 - 8$
  This simplifies to:
  $-2x = 0$

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

  $x = \frac{0}{-2} = 0$

Therefore, the solution to the equation is $x = 0$.

To solve the equation $x + 7 = 14$, we aim to find the value of $x$ by isolating it on one side.

• Step 1: Identify the current equation: $x + 7 = 14$.
• Step 2: To isolate $x$, perform the inverse operation. Subtract 7 from both sides to maintain equality.
• Step 3: Simplify both sides: $x + 7 - 7 = 14 - 7$.
• Step 4: This simplifies to $x = 7$.

Therefore, we have found that the solution to the equation $x + 7 = 14$ is $x = 7$, which matches the given answer choice 2.

To solve the equation $5x = 25$, we will isolate $x$ using division:

• Divide both sides of the equation by 5:

After performing the division, we get:

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

To solve for $x$, start by isolating $x$ on one side of the equation:
Subtract 8 from both sides:
$x + 8 - 8 = 10 - 8$ simplifies to
$x = 2$.

To solve the equation $\frac{1}{3}x = 9$, we need to isolate the variable $x$. To accomplish this, we can multiply both sides of the equation by 3, the reciprocal of $\frac{1}{3}$.

Step-by-step solution:

• Step 1: Multiply both sides by 3.
  $\left(3 \times \frac{1}{3}\right)x = 3 \times 9$
• Step 2: Simplify the left side.
  This gives us $1x = 27$, since $\left(3 \times \frac{1}{3}\right) = 1$.
• Step 3: Conclude that $x = 27$.

Therefore, the solution to the equation is $x = 27$. This matches choice number 1 from the provided options.

To solve the equation $x + 5 = 8$, follow these steps:

• Step 1: Start with the original equation:
  $x + 5 = 8$.
• Step 2: Subtract 5 from both sides of the equation to isolate $x$:
  $x + 5 - 5 = 8 - 5$.
• Step 3: Simplify both sides:
  $x = 3$.

Therefore, the solution to the equation is $x = 3$.

The correct answer choice is: :

3

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 for $x$, start by isolating $x$ on one side of the equation:
Subtract 7 from both sides:
$x + 7 - 7 = 12 - 7$ simplifies to
$x = 5$.

To solve the equation $5 - x = 4$, we aim to isolate $x$ on one side of the equation.

We start by considering the equation:
$5 - x = 4$

Step 1: Eliminate 5 from the left side to isolate terms involving $x$. To do this, subtract 5 from both sides of the equation:

$(5 - x) - 5 = 4 - 5$

Step 2: Simplify both sides:

$-x = -1$

Step 3: To solve for $x$, multiply or divide both sides by $-1$ to change the sign of $x$:

$-1 \cdot -x = -1 \cdot -1$

This simplifies to:

$x = 1$

Therefore, the solution to the equation $5 - x = 4$ is $x = 1$.

The correct answer is $x = 1$.

To open parentheses we will use the formula:

$a(x+b)=ax+ab$

$(7\times-2x)+(7\times5)=77$

We multiply accordingly

$-14x+35=77$

We will move the 35 to the right section and change the sign accordingly:

$-14x=77-35$

We solve the subtraction exercise on the right side and we will obtain:

$-14x=42$

We divide both sections by -14

$\frac{-14x}{-14}=\frac{42}{-14}$

$x=-3$

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

• Step 1: Combine like terms of the given equation.
• Step 2: Solve for the variable $m$.

Now, let's work through these steps:

Step 1: Combine like terms:
We start with the equation $7m + 3m - 40m = 0$.
Combining these like terms entails adding or subtracting the coefficients of $m$:

$(7 + 3 - 40)m = 0$
Calculate the sum and difference of these coefficients:
$(10 - 40)m = 0$

This simplifies to:
$-30m = 0$

Step 2: Solve for $m$:
To isolate $m$, divide both sides by $-30$:
$m = \frac{0}{-30}$

Calculate the right-hand side:

$m = 0$

Therefore, the solution to the problem is $m = 0$. This corresponds to choice 3 from the provided answer options.

To solve for $x$, start by isolating $x$ on one side of the equation:
Subtract 3 from both sides:
$x + 3 - 3 = 7 - 3$ simplifies to
$x = 4$.

To solve this problem, we will follow these steps:

• Step 1: Identify the given equation $3 + x = 4$.
• Step 2: Use subtraction to isolate the variable $x$.

Now, let's work through these steps:
Step 1: We have the equation: $3 + x = 4$.
Step 2: Subtract 3 from both sides of the equation to isolate $x$:

$3 + x - 3 = 4 - 3$

This simplifies to:

$x = 1$

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

Step-by-step solution:

1. Begin with the equation: $x + 9 = 15$

2. Subtract 9 from both sides: $x + 9 - 9 = 15 - 9$, which simplifies to $x = 6$

To solve for $x$ in the equation $6x = 72$, follow these steps:

Step 1: Identify the equation and the coefficient of $x$.
The given equation is $6x = 72$, where the coefficient of $x$ is 6.

Step 2: Isolate $x$ by dividing both sides of the equation by the coefficient (6).
Perform the division: $x = \frac{72}{6}$.

Step 3: Simplify the result.
Calculating $\frac{72}{6}$, we get $x = 12$.

Therefore, the solution to the equation is $x = 12$.