Solving Quadratic Equations using Factoring: Solving the problem

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

• Step 1: Eliminate the fraction by multiplying both sides of the equation by 2.
• Step 2: Simplify the resulting equation.
• Step 3: Solve for $x$ by adding 3 to both sides.

Now, let's work through each step:
Step 1: Starting with the equation $2 = (x-3) \times \frac{1}{2}$, we multiply both sides by 2 to get rid of the fraction:
$2 \times 2 = (x-3) \times \frac{1}{2} \times 2$

This simplifies to:
$4 = x - 3$

Step 2: Next, we solve for $x$ by adding 3 to both sides:
$4 + 3 = x$

Therefore, $x = 7$.

Therefore, the solution to the problem is $x = 7$.

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

• Step 1: Eliminate the fraction in the given equation by multiplying through by 3.
• Step 2: Solve for $x$ by isolating it on one side of the equation.
• Step 3: Compare the result to the provided answer choices and select the correct one.

Now, let's work through each step:

Step 1: Begin with the given equation:
$3 = (3x - 1) \times \frac{1}{3}$

To eliminate the fraction, multiply both sides by 3:
$3 \times 3 = (3x - 1) \times \frac{1}{3} \times 3$

This simplifies to:
$9 = 3x - 1$

Step 2: Solve for $x$ by isolating it:

Add 1 to both sides to remove the constant term on the right side:
$9 + 1 = 3x - 1 + 1$

Thus, we have:
$10 = 3x$

Finally, divide both sides by 3 to isolate $x$:
$x = \frac{10}{3}$

Step 3: Compare the result to the provided answer choices:

The value $x = \frac{10}{3}$ is equivalent to the mixed number representation $3 \frac{1}{3}$.

Therefore, the solution to the problem is $3\frac{1}{3}$, which matches choice 3.

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

• Step 1: Clear the fraction by multiplying both sides of the equation by 2.
• Step 2: Isolate $x$ by adding 4 to both sides of the resulting equation.

Now, let's work through each step:
Step 1: Multiply both sides of the equation by 2 to eliminate the fraction:
$7 \times 2 = (x - 4) \times \frac{1}{2} \times 2$

This simplifies to:
$14 = x - 4$

Step 2: Solve for $x$ by adding 4 to both sides:
$14 + 4 = x - 4 + 4$

This simplifies to:
$x = 18$

Therefore, the solution to the equation is $x = 18$.

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

• Step 1: Eliminate the fraction by multiplying both sides of the equation by 4.
• Step 2: Simplify and solve for $x$.

Let's work through each step:

Step 1: The original equation is $2 = (x-2) \times \frac{1}{4}$.
To eliminate the fraction, multiply both sides by 4:
$4 \times 2 = 4 \times (x-2) \times \frac{1}{4}$

This simplifies to:
$8 = x - 2$

Step 2: Solve for $x$.
Add 2 to both sides to isolate $x$:
$8 + 2 = x$

This simplifies to:
$x = 10$

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

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

• Step 1: Eliminate the fraction by multiplying both sides of the equation by the reciprocal of $\frac{2}{3}$.
• Step 2: Simplify the resulting equation.
• Step 3: Solve for $x$.

Now, let's work through each step:

Step 1: Multiply both sides of the equation by $\frac{3}{2}$ to eliminate the fraction. This gives:

$\frac{3}{2} \times 4 = \frac{3}{2} \times \left((2x - 1) \times \frac{2}{3}\right)$

Simplifying, the left side becomes:

$6 = 2x - 1$

Step 2: Add 1 to both sides to solve for $2x$:

$6 + 1 = 2x - 1 + 1$

$7 = 2x$

Step 3: Divide both sides by 2 to isolate $x$:

$\frac{7}{2} = x$

Converting $\frac{7}{2}$ to a decimal gives us:

$x = 3.5$

Therefore, the solution to the problem is $x = 3.5$.

To solve this problem, we'll perform the following calculations:

• Step 1: Multiply both sides of the equation by $\frac{5}{2}$ to eliminate the fraction:

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

This simplifies to:

$\frac{25}{2} = 2x - 2$

• Step 2: Solve for $x$ by isolating it on one side of the equation.
• Add 2 to both sides to shift the constant:

$\frac{25}{2} + 2 = 2x - 2 + 2$

$\frac{25}{2} + \frac{4}{2} = 2x$

$\frac{29}{2} = 2x$

• Step 3: Solve for $x$ by dividing both sides by 2:

$x = \frac{29}{4}$

Therefore, the solution to the problem is that $x = 7\frac{1}{4}$.

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

• Step 1: Eliminate the fraction by multiplying both sides of the equation by 8.
• Step 2: Simplify the resulting equation.
• Step 3: Solve for $x$.

Let's work through each step:

Step 1: Multiply both sides of the equation by 8 to eliminate the fraction:

$4 \times 8 = (2x - 4) \times \frac{1}{8} \times 8$

This simplifies to:

$32 = 2x - 4$

Step 2: Isolate $2x$ by adding 4 to both sides:

$32 + 4 = 2x$ $36 = 2x$

Step 3: Solve for $x$ by dividing both sides by 2:

$\frac{36}{2} = x$ $18 = x$

Therefore, the solution to the problem is: $x = 18$.

Comparing this with the given choices, the correct choice is:

• Choice 3: $(18)$