Solving an Equation by Multiplication/ Division: Exercises on Both Sides (of the Equation)

First, we arrange the two sections so that the right side contains the values with the coefficient x and the left side the numbers without the x

Let's remember to maintain the plus and minus signs accordingly when we move terms between the sections.

First, we move a$5x$ to the right section and then the 22 to the left side. We obtain the following equation:

$-8-22=10x-5x$

We subtract both sides accordingly and obtain the following equation:

$-30=5x$

We divide both sections by 5 and obtain:

$-6=x$

To solve the equation $6 - 7x = -5x + 8$, we will follow these steps:

• Step 1: Move all terms involving $x$ to one side of the equation by adding $5x$ to both sides.
• Step 2: Simplify both sides of the equation.
• Step 3: Solve for $x$.

Now, let's work through each step:

Step 1: Add $5x$ to both sides to move the $x$-term to the left side:

$6 - 7x + 5x = 8$

Step 2: Simplify the equation by combining the $x$-terms on the left side:

$6 - 2x = 8$

Step 3: To isolate $-2x$ on the left, subtract $6$ from both sides:

$-2x = 8 - 6$

Simplify the right side:

$-2x = 2$

Finally, divide both sides by $-2$ to solve for $x$:

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

$x = -1$

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

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

To solve the equation $\frac{1}{3}(x+9) = 4+\frac{2}{3}x$, we will follow these steps:

• Step 1: Clear fractions by multiplying through by the least common denominator.
• Step 2: Simplify the equation to combine like terms.
• Step 3: Solve for $x$.

Let's begin:

Step 1: Multiply every term in the equation by 3 to eliminate fractions:

This simplifies to:

Step 2: Rearrange the equation to get all $x$ terms on one side and constant terms on the other:

Subtract $2x$ from both sides:

Which simplifies to:

Next, subtract 9 from both sides to isolate terms involving $x$:

Step 3: Solve for $x$ by multiplying both sides by -1:

Thus, the solution to the equation is $x = -3$.

First, let's isolate a from the parentheses in the equation on the right side. We'll remember that minus times minus becomes plus, so we get the equation:

$a^4+7a-5=2a+a^4+3a+a$

Let's continue solving the equation on the right side by adding $2a+3a+a=5a+a=6a$

Now the equation we got is:

$a^4+7a-5=6a+a^4$

Let's divide both sides by $a^4$ and we get:

$7a-5=6a$

Now let's move 6a to the left side and the number 5 to the right side, remembering to change the plus and minus signs accordingly.

The equation we got now is:

$7a-6a=5$

Let's solve the subtraction and we get:

$1a=5$

Let's divide both sides by 1 and we find that $a=5$

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

• Step 1: Distribute on the right-hand side
• Step 2: Combine like terms on the left-hand side
• Step 3: Isolate the variable $x$
• Step 4: Solve for $x$

Now, let's work through each step:

Step 1: Distribute on the right-hand side of the equation:

$2x + 45 - \frac{1}{3}x = 5(x + 7) \quad \Rightarrow \quad 2x + 45 - \frac{1}{3}x = 5x + 35$

Step 2: Combine like terms on the left-hand side:

Combine $2x$ and $-\frac{1}{3}x$ on the left:

$2x - \frac{1}{3}x = \frac{6}{3}x - \frac{1}{3}x = \frac{5}{3}x$

The equation becomes:

$\frac{5}{3}x + 45 = 5x + 35$

Step 3: Move all terms with $x$ to one side and constants to the other:

$\frac{5}{3}x - 5x = 35 - 45$

Step 4: Simplify and solve for $x$:

$\frac{5}{3}x - \frac{15}{3}x = -10$ $-\frac{10}{3}x = -10$

Step 5: Solve for $x$ by dividing both sides by $-\frac{10}{3}$:

$x = \frac{-10}{-\frac{10}{3}} = 3$

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

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

• Step 1: Expand and simplify the right-hand side.
• Step 2: Set the equation to zero by moving all terms to one side.
• Step 3: Simplify to obtain a standard quadratic equation.
• Step 4: Use the quadratic formula to find the possible solutions for $x$.

Now, let's work through each step:

Step 1:
Expand the right-hand side:
$(-x + 7)(4x - 9) = -x(4x) - x(-9) + 7(4x) - 7(9)$
= $-4x^2 + 9x + 28x - 63$
Considering both sides: $-4(x^2 + 5) = -4x^2 + 9x + 28x - 63 + 5$.

Step 2:
Simplify further by calculating:
$-4x^2 - 20 = -4x^2 + 37x - 58$.

Step 3:
Move all terms to one side to achieve zero on the right-hand side:
$-4x^2 - 20 + 4x^2 - 37x + 58 = 0$
Simplifying, we get: $37x + 38 = 0$.

Step 4:
Since the $x^2$ terms cancel, it's actually a linear equation:
$37x = -38$.
Solving for $x$, we divide both sides by 37:
$x = \frac{-38}{37} = -1\frac{1}{37}$.

Therefore, the solution to the problem is $x = 1\frac{1}{37}$.