Solving an Equation by Multiplication/ Division: Solving an equation with fractions

The goal is to solve the equation $4 = 3y$ to find the value of $y$. To do this, we can follow these steps:

• Step 1: Divide both sides of the equation by 3 to isolate $y$.
• Step 2: Simplify the result to solve for $y$.

Now, let's work through the solution:

Step 1: We start with the equation:

$4 = 3y$

To solve for $y$, divide both sides by 3:

$y = \frac{4}{3}$

Step 2: Simplify the fraction:

$y = \frac{4}{3} = 1 \frac{1}{3}$

Therefore, the solution to the equation is $y = 1 \frac{1}{3}$.

This corresponds to choice $y = 1\frac{1}{3}$ in the provided multiple-choice answers.

To solve the equation $6x = -12.6$, we need to isolate $x$. We achieve this by performing the following steps:

• Step 1: Identify the coefficient of $x$, which is $6$. We aim to isolate $x$ by dividing both sides of the equation by $6$.
• Step 2: Divide both sides of the equation by $6$ to solve for $x$. This will eliminate the coefficient and reveal the value of $x$.

Let's perform these calculations:

$\frac{6x}{6} = \frac{-12.6}{6}$

Simplifying both sides gives:

$x = -2.1$

Therefore, the solution to the equation is $\boldsymbol{x = -2.1}$.

We begin by multiplying the simple fraction by y:

$\frac{-1}{5}\times y=-25$

We then reduce both terms by $-\frac{1}{5}$

$y=\frac{-25}{-\frac{1}{5}}$

Finally we multiply the fraction by negative 5:

$y=-25\times(-5)=125$

To solve the equation $3b = \frac{7}{6}$ for the variable $b$, we will perform the following steps:

• Step 1: Identify the equation. The given equation is $3b = \frac{7}{6}$.
• Step 2: Isolate the variable. Divide both sides by 3 to solve for $b$.

When we divide both sides of the equation by 3, we obtain:

Step 3: Simplify the expression. Dividing a fraction by an integer is equivalent to multiplying the denominator of the fraction by that integer:

The denominator becomes:

Thus, the solution to the equation is $b = \frac{7}{18}$.

This matches the correct answer choice among the given options.

Therefore, the value of $b$ is $b = \frac{7}{18}$.

To solve the equation $\frac{3x}{4} = 16$, we will eliminate the fraction by multiplying both sides by 4.

• Step 1: Multiply both sides by 4:
  $\left(\frac{3x}{4}\right) \times 4 = 16 \times 4$
• Step 2: Simplify:
  $3x = 64$
• Step 3: Solve for $x$ by dividing both sides by 3:
  $x = \frac{64}{3}$
• Step 4: Simplify the fraction to a mixed number:
  $x = 21\frac{1}{3}$

Therefore, the solution to the equation $\frac{3x}{4} = 16$ is $x = 21\frac{1}{3}$.

To solve this problem, follow these steps:

• Step 1: Identify the equation and its components. The equation is $4x - 6.9 = 2.2x + 5$.
• Step 2: Move the terms involving $x$ to one side of the equation. Subtract $2.2x$ from both sides:
  $4x - 2.2x - 6.9 = 5$.
• Step 3: Combine like terms on the left side:
  $1.8x - 6.9 = 5$.
• Step 4: Move constant terms to the other side by adding $6.9$ to both sides:
  $1.8x = 5 + 6.9$.
• Step 5: Simplify the equation on the right side:
  $1.8x = 11.9$.
• Step 6: Solve for $x$ by dividing both sides by $1.8$:
  $x = \frac{11.9}{1.8}$.
• Step 7: Simplify the fraction if possible:
  Converting $\frac{11.9}{1.8}$ to a mixed number, $x = 6\frac{11}{18}$.

Therefore, the solution to the equation is $x = 6\frac{11}{18}$.

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

• Step 1: Start with the given equation.
• Step 2: Use cross-multiplication to eliminate the fractions.
• Step 3: Simplify the resulting expression.
• Step 4: Solve for the variable $a$.

Now, let's work through each step:
Step 1: The equation given is $\frac{a}{6} = \frac{6}{7}$.
Step 2: We apply cross-multiplication: Multiply both sides to get $a \times 7 = 6 \times 6$.
Step 3: Simplify the equation: $7a = 36$.
Step 4: Solve for $a$ by dividing both sides by 7:
$a = \frac{36}{7}$.
This fraction can be converted to a mixed number: $a = 5\frac{1}{7}$.

Therefore, the solution to the problem is $a = 5\frac{1}{7}$.

To solve $\frac{x-5}{7} = \frac{2}{11}$, we will use cross-multiplication:

• Step 1: Cross-multiply the equation: $(x-5) \times 11 = 7 \times 2$.
• Step 2: Simplify both sides: $11(x - 5) = 14$.
• Step 3: Distribute the 11 on the left side: $11x - 55 = 14$.
• Step 4: Add 55 to both sides to isolate the $11x$ term: $11x = 14 + 55$.
• Step 5: Calculate the right side: $11x = 69$.
• Step 6: Divide both sides by 11 to solve for $x$: $x = \frac{69}{11}$.

Therefore, the solution to the problem is $x = \frac{69}{11}$, which matches the first answer choice provided.

To solve the equation $\frac{x+2}{3}=\frac{4}{5}$, we can follow the method of cross-multiplication:

• Step 1: Cross-multiply to eliminate the fractions, giving us:

$(x + 2) \cdot 5 = 4 \cdot 3$

• Step 2: Simplify both sides of the equation:

$5(x + 2) = 12$

• Step 3: Distribute the 5 on the left side:

$5x + 10 = 12$

• Step 4: Subtract 10 from both sides to isolate the term with $x$:

$5x = 2$

• Step 5: Divide both sides by 5 to solve for $x$:

$x = \frac{2}{5}$

Therefore, the solution to the equation is $\frac{2}{5}$.

To solve the equation $\frac{x-4}{18} = \frac{7}{9}$, we'll follow these steps:

• Step 1: Apply the principle of cross-multiplication to eliminate fractions.

• Step 2: Solve for the linear expression in terms of $x$.

• Step 3: Isolate $x$ and solve the equation completely.

Now, let's work through each step:
Step 1: Cross-multiply to eliminate the fractions. The equation becomes:

$(x-4) \cdot 9 = 18 \cdot 7 \\ 9(x-4) = 126$

Step 2: Distribute the 9 on the left-hand side:

$9x - 36 = 126$

Step 3: Add 36 to both sides to isolate the term with $x$:

$9x = 126 + 36 9x = 162$

Step 4: Divide both sides by 9 to solve for $x$:

$x = \frac{162}{9} \\ x = 18$

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

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

• Step 1: Convert the mixed number to an improper fraction
• Step 2: Isolate $b$ using multiplication
• Step 3: Simplify to find the value of $b$

Now, let's work through each step:

Step 1: Convert $4\frac{1}{2}$ to an improper fraction:

$4\frac{1}{2} = \frac{9}{2}$

Step 2: Isolate $b$ on one side of the equation:

The equation becomes $70 = \frac{9}{2}b$

To isolate $b$, multiply both sides by the reciprocal of $\frac{9}{2}$:

$b = 70 \times \frac{2}{9}$

Step 3: Perform the multiplication:

$b = \frac{70 \times 2}{9}$

$b = \frac{140}{9}$

The improper fraction $\frac{140}{9}$ converts to a mixed number:

$b = 15 \frac{5}{9}$

Therefore, the solution to the problem is $b = 15\frac{5}{9}$.

First, we cross multiply:

$8\times(x+4)=3\times7$

We multiply the right section and expand the parenthesis, multiplying each of the terms by 8:

$8x+32=21$

We rearrange the equation remembering change the plus and minus signs accordingly:

$8x=21-32$
Solve the subtraction exercise on the right side and divide by 8:

$8x=-11$

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

Convert the simple fraction into a mixed fraction:

$x=-1\frac{3}{8}$

To solve the equation $\frac{5}{x-8} = \frac{3}{4x}$ for the variable $x$, we will follow these steps:

Step 1: Apply cross-multiplication to the equation. This involves multiplying the numerator of each fraction by the denominator of the other fraction:

Step 2: Simplify both sides of the resulting equation:

Step 3: Rearrange the equation to isolate terms involving $x$ on one side:

This simplifies to:

Step 4: Solve for $x$ by dividing both sides of the equation by 17:

Therefore, the solution to the equation is:

$x = \frac{-24}{17}$

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

• Step 1: Identify that the given equation is $\frac{5}{8-x} = \frac{3}{2x}$.
• Step 2: Cross-multiply to eliminate the fractions.
• Step 3: Solve the resulting linear equation.
• Step 4: Check for any restrictions on $x$.

Now, let's work through each step:

Step 1: We have the equation:

$\frac{5}{8-x} = \frac{3}{2x}$

Step 2: Cross-multiply to get:

$5 \cdot 2x = 3 \cdot (8-x)$

This simplifies to:

$10x = 24 - 3x$

Step 3: Solve for $x$ by isolating it on one side of the equation. Add $3x$ to both sides:

$10x + 3x = 24$

This simplifies to:

$13x = 24$

Now, divide both sides by 13:

$x = \frac{24}{13}$

Step 4: Verify that this value does not make any of the original denominators zero. For $x = \frac{24}{13}$, the terms $8-x$ and $2x$ are well-defined, and neither is zero:

$8 - \frac{24}{13} = \frac{80}{13} - \frac{24}{13} = \frac{56}{13} \neq 0$

$2 \times \frac{24}{13} = \frac{48}{13} \neq 0$

No issues arise from substituting back, so our solution is valid.

Therefore, the solution to the problem is $x = \frac{24}{13}$, which corresponds to choice 3.

To solve this problem, we'll use a step-by-step approach:

Step 1: Set up the equation based on the problem statement.
The total cost Lionel pays is given by the formula:

$4.5x = 45$

Here, $x$ is the number of packs Lionel buys, and $4.5 is the cost per pack.

Step 2: Solve for $x$.
To find $x$, divide both sides of the equation by 4.5:

$x = \frac{45}{4.5}$

Step 3: Perform the division.
Carrying out the division,

$x = \frac{45}{4.5} = 10$

Therefore, Lionel buys $x = 10$ packs of paper.

To get rid of the fraction mechanics, we will cross multiply between the sides:

$4(8x-4)=5(2x+2)$

We expand the parentheses by multiplying the outer element by each of the elements inside the parentheses:

$32x-16=10x+10$

We arrange the sides accordingly so that the elements with the X are on the left side and those without the X are on the right side:

$32x-10x=10+16$

We calculate the elements:

$22x=26$

We divide the two sections by 22:

$\frac{22x}{22}=\frac{26}{22}$

$x=\frac{26}{22}$

To solve for $x$ in the equation $\frac{-8+x}{3}=\frac{x+4}{9}$, we'll follow these steps:

• Step 1: Eliminate the fractions by finding a common denominator.
• Step 2: Simplify the resulting equation.
• Step 3: Isolate $x$ to solve the equation.

Let's proceed step by step:

Step 1: The equation $\frac{-8+x}{3}=\frac{x+4}{9}$ contains denominators 3 and 9. The least common denominator (LCD) is 9. To eliminate the fractions, multiply every term of the equation by 9.

$9 \times \frac{-8+x}{3} = 9 \times \frac{x+4}{9}$

Simplifying, we have:

$3(-8 + x) = x + 4$

Step 2: Distribute the 3 on the left side:

$3 \times -8 + 3 \times x = x + 4$

This simplifies to:

$-24 + 3x = x + 4$

Step 3: Isolate $x$. First, subtract $x$ from both sides of the equation:

$-24 + 3x - x = x + 4 - x$

This simplifies to:

$-24 + 2x = 4$

Next, add 24 to both sides to further isolate $x$:

$-24 + 24 + 2x = 4 + 24$

This simplifies to:

$2x = 28$

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

$x = \frac{28}{2}$

Simplifying this gives:

$x = 14$

Therefore, the solution to the equation is $x = 14$.

To solve the equation $\frac{5-x}{8} = \frac{3+x}{2}$, we will follow these steps:

• Step 1: Eliminate the fractions by cross-multiplying.
• Step 2: Simplify the resulting equation.
• Step 3: Solve for $x$.

Let's proceed with each step:

Step 1: Cross-multiply.
Cross-multiplying gives:

$(5 - x) \times 2 = (3 + x) \times 8$

Which simplifies to:

$2(5 - x) = 8(3 + x)$

Step 2: Expand and simplify.
Distribute the constants inside the parentheses:

$10 - 2x = 24 + 8x$

Step 3: Isolate $x$.
Add $2x$ to both sides to bring all terms involving $x$ to one side:

$10 = 24 + 10x$

Subtract 24 from both sides to isolate the term with $x$:

$10 - 24 = 10x$

Simplify:

$-14 = 10x$

Finally, divide both sides by 10 to solve for $x$:

$x = \frac{-14}{10}$

Therefore, the solution to the problem is $x = \frac{-14}{10}$, which corresponds to choice .

To solve for the price per kg of cucumbers, follow these steps:

• Step 1: Determine the total cost of tomatoes.
  Given that the cost of tomatoes is 2.8 per kg, and Maggie buys 2 kg, the cost of tomatoes is calculated as:
  \( 2 \, \text{kg} \times 2.8 \, \text{dollars/kg} = 5.6 \, \text{dollars} .

• Step 2: Write the equation for total cost.
  The total cost for tomatoes and cucumbers combined is given as 7.1. Let \( x represent the cost per kg of cucumbers. The equation representing the total cost is:
  $5.6 + 0.6x = 7.1$.

• Step 3: Solve the equation for $x$.
  Subtract the cost of tomatoes from both sides of the equation to find the cost of cucumbers:
  $0.6x = 7.1 - 5.6$.
  Simplifying the right side gives:
  $0.6x = 1.5$.

• Step 4: Isolate $x$ by dividing both sides by 0.6:
  $x = \frac{1.5}{0.6}$.
  Simplify the division to find $x$:
  $x = 2.5$.

Therefore, the value per kg of cucumbers is $x = 2.5$ dollars.