Solving Equations by using Addition/ Subtraction: Solving an equation with fractions

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

• Step 1: Convert the mixed number $2\frac{1}{2}$ to an improper fraction.
• Step 2: Subtract $2\frac{1}{2}$ from both sides of the equation to isolate $a$.
• Step 3: Simplify the result to find the value of $a$.

Now, let's work through each step:

Step 1: Convert $2\frac{1}{2}$ to an improper fraction. $2\frac{1}{2} = \frac{5}{2}$.

Step 2: The equation becomes $a + \frac{5}{2} = 4$. To isolate $a$, subtract $\frac{5}{2}$ from both sides:

$a = 4 - \frac{5}{2}$

Step 3: Convert 4 into a fraction with the same denominator to perform the subtraction. $4 = \frac{8}{2}$.

$a = \frac{8}{2} - \frac{5}{2} = \frac{3}{2}$.

The improper fraction $\frac{3}{2}$ can be converted back to a mixed number, giving $a = 1\frac{1}{2}$.

Therefore, the solution to the problem is $a = 1\frac{1}{2}$.

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

• Step 1: Convert mixed numbers to improper fractions
• Step 2: Isolate the variable $b$ by applying algebraic operations
• Step 3: Simplify the result to find the value of $b$

Now, let's work through each step:

Step 1: Convert the mixed numbers to improper fractions.
$3\frac{4}{5} = \frac{15}{5} + \frac{4}{5} = \frac{19}{5}$
$-8\frac{3}{5} = -\left(\frac{40}{5} + \frac{3}{5}\right) = -\frac{43}{5}$

Step 2: Add $\frac{19}{5}$ to both sides of the equation to isolate $b$.

$b - \frac{19}{5} = -\frac{43}{5}$

Add $\frac{19}{5}$ to both sides:
$b - \frac{19}{5} + \frac{19}{5} = -\frac{43}{5} + \frac{19}{5}$
This simplifies to:
$b = \frac{-43+19}{5} = \frac{-24}{5} = -4\frac{4}{5}$

Therefore, the solution to the problem is $b = -4\frac{4}{5}$.

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

• Step 1: Find a common denominator for the fractions involved.
• Step 2: Simplify the left-hand side of the equation.
• Step 3: Solve the resulting equation for $x$.

Now, let's work through each step:

Step 1: The given equation is $\frac{1}{3}x - \frac{1}{2}x = 3$. The fractions $\frac{1}{3}$ and $\frac{1}{2}$ need a common denominator. The least common denominator for 3 and 2 is 6.

Step 2: Convert each fraction: $\frac{1}{3} = \frac{2}{6} \quad \text{and} \quad \frac{1}{2} = \frac{3}{6}$ Thus, the equation $\frac{1}{3}x - \frac{1}{2}x = 3$ becomes $\frac{2}{6}x - \frac{3}{6}x = 3$ Combine the terms: $\left(\frac{2}{6} - \frac{3}{6}\right)x = -\frac{1}{6}x$ So the equation is: $-\frac{1}{6}x = 3$

Step 3: Solve for $x$: To isolate $x$, multiply both sides of the equation by $-6$ (the reciprocal of $-\frac{1}{6}$): $x = 3 \times (-6)$ $x = -18$

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

To solve the equation $\frac{2}{5}x + \frac{3}{4}x = 1$, we will follow these steps:

• Step 1: Find a common denominator to combine the fractions on the left-hand side.
• Step 2: Simplify the equation.
• Step 3: Isolate the variable $x$.

Now, let's work through each step:
Step 1: The denominators are 5 and 4. The least common denominator (LCD) is 20.
Convert each term: $\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}$ and $\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$.

Step 2: Combine the fractions: $\frac{8}{20}x + \frac{15}{20}x = \frac{23}{20}x$.
The equation now is $\frac{23}{20}x = 1$.

Step 3: Solve for $x$ by multiplying both sides by the reciprocal of $\frac{23}{20}$, which is $\frac{20}{23}$.
Thus, $x = 1 \times \frac{20}{23} = \frac{20}{23}$.

Therefore, the solution to the equation is $\boxed{\frac{20}{23}}$.

To solve the equation $\frac{1}{8}x + \frac{2}{3}x = 3$, we need to clear the fractions by finding the least common denominator.

Step 1: Identify the least common denominator of the fractions.
The denominators are 8 and 3. The least common denominator (LCD) is 24.

Step 2: Rewrite the equation with the LCD to get rid of the fractions:
Multiply each term by 24:
$24 \times \frac{1}{8} x + 24 \times \frac{2}{3} x = 24 \times 3$.

Step 3: Simplify each term:
$\frac{24}{8}x = 3x$,
$\frac{24}{3}x = 16x$,
Thus, the equation becomes $3x + 16x = 72$.

Step 4: Combine like terms:
$19x = 72$.

Step 5: Solve for $x$ by dividing both sides by 19:
$x = \frac{72}{19}$.

Convert the improper fraction to a mixed number:
Divide 72 by 19, which gives 3 with a remainder of 15. Thus, $x = 3\frac{15}{19}$.

Therefore, the solution to the problem is $x = 3\frac{15}{19}$.

To solve the provided equation $\frac{2}{7}x - \frac{2}{3}x = 4$, we need to combine like terms on the left-hand side. Let's work step-by-step:

• Step 1: Identify the common denominator of the fractions $\frac{2}{7}$ and $\frac{2}{3}$.
  The least common denominator of 7 and 3 is 21.

• Step 2: Rewrite each term with the common denominator of 21:
  $\frac{2}{7}x = \frac{2 \times 3}{7 \times 3}x = \frac{6}{21}x$ and
  $\frac{2}{3}x = \frac{2 \times 7}{3 \times 7}x = \frac{14}{21}x$.

• Step 3: Combine these like terms:
  $\frac{6}{21}x - \frac{14}{21}x = \frac{6 - 14}{21}x = \frac{-8}{21}x$.

• Step 4: Rewrite the equation with the combined terms:
  $\frac{-8}{21}x = 4$.

• Step 5: Solve for $x$ by multiplying both sides by the reciprocal of $-\frac{8}{21}$:
  $x = 4 \times \frac{21}{-8} = 4 \times -\frac{21}{8} = -\frac{84}{8} = -10.5$ or $-10\frac{1}{2}$.

Therefore, the solution to the equation is $x = -10\frac{1}{2}$.

Solve the equation by following these steps:

• Step 1: Find the least common multiple (LCM) of the denominators $4$, $3$, and $8$. The LCM is $24$.
• Step 2: Rewrite each term so that the denominator is $24$.
  $\frac{1}{4}x = \frac{6}{24}x$
  $-\frac{2}{3}x = -\frac{16}{24}x$
  $\frac{1}{8}x = \frac{3}{24}x$
• Step 3: Combine the terms over the common denominator:
  $\left(\frac{6}{24} - \frac{16}{24} + \frac{3}{24}\right)x = 5$
• Step 4: Simplify the left side:
  $\frac{6 - 16 + 3}{24}x = 5$
  $\frac{-7}{24}x = 5$
• Step 5: Solve for $x$ by isolating it:
  Multiply both sides by $-24/7$:
  $x = 5 \times \left(-\frac{24}{7}\right)$
• Step 6: Perform the multiplication:
  $x = -\frac{120}{7}$
• Step 7: Convert the improper fraction to a mixed number:
  The mixed number is $-17\frac{1}{7}$.

The solution to the equation is $x = -17\frac{1}{7}$.

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

• Step 1: Identify the common denominator.
• Step 2: Rewrite the equation with the common denominator.
• Step 3: Combine like terms.
• Step 4: Solve the simplified equation for $x$.

Now, let's work through each step:

Step 1: The denominators of the fractions are 4, 3, and 6. The least common denominator is 12.

Step 2: Rewrite each term with the common denominator:

• $-\frac{1}{4}x = -\frac{3}{12}x$
• $\frac{2}{3}x = \frac{8}{12}x$
• $-\frac{1}{6}x = -\frac{2}{12}x$

Step 3: Combine the fractions:

$-\frac{3}{12}x + \frac{8}{12}x - \frac{2}{12}x = 2$

This simplifies to $\frac{3}{12}x = 2$.

Step 4: Simplify and solve for $x$:

Multiply both sides by 12 to clear the denominator:

$3x = 24$

Divide both sides by 3:

$x = 8$

Therefore, the solution to the problem is $x = 8$.

The common denominator of 8, 3, and 5 is 120.

Now we multiply each numerator by the corresponding number to reach 120 and thus cancel the fractions and obtain the following equation:

$(1\times x\times15)-(2\times x\times40)+(1\times x\times24)=1\times120$

We multiply the exercises in parentheses accordingly:

$15x-80x+24x=120$

We will solve the left side (from left to right) and will obtain:

$(15x-80x)+24x=120$

$-65x+24x=120$

$-41x=120$

We reduce both sides by $-41$

$\frac{-41x}{-41}=\frac{120}{-41}$

We find that x is equal$x=-\frac{120}{41}$

To solve this equation, we will follow these steps:

• Step 1: Identify the least common denominator (LCD) for the fractions $\frac{2}{4}x$, $\frac{1}{2}x$, $\frac{3}{8}x$, and $\frac{1}{5}x$.
• Step 2: Convert each fraction to have the LCD as the denominator.
• Step 3: Combine the fractions to form a single expression.
• Step 4: Solve the resulting equation for $x$.

Now, let's work through each step:

Step 1: The denominators are 4, 2, 8, and 5. The LCD of these numbers is 40.

Step 2: Convert each fraction:

$\frac{2}{4}x = \frac{2 \times 10}{40}x = \frac{20}{40}x$

$\frac{1}{2}x = \frac{1 \times 20}{40}x = \frac{20}{40}x$

$\frac{3}{8}x = \frac{3 \times 5}{40}x = \frac{15}{40}x$

$-\frac{1}{5}x = -\frac{1 \times 8}{40}x = -\frac{8}{40}x$

Step 3: Combine all fractions:

$\frac{20}{40}x + \frac{20}{40}x + \frac{15}{40}x - \frac{8}{40}x$

$= \frac{20 + 20 + 15 - 8}{40}x$

$= \frac{47}{40}x$

Step 4: Solve the equation $\frac{47}{40}x = 1$.

Multiply both sides by $\frac{40}{47}$ to solve for $x$:

$x = 1 \times \frac{40}{47}$

$x = \frac{40}{47}$

Therefore, the solution to the problem is $x = \frac{40}{47}$.

To solve the given equation $\frac{1}{4}x - \frac{1}{5}x + \frac{2}{3}x - \frac{2}{5}x = 1$, follow these steps:

Step 1: Find a common denominator for the fractions involved. The denominators are 4, 5, 3, and again 5. The least common multiple (LCM) of these numbers is 60.

Step 2: Rewrite each fraction with the common denominator of 60:

• $\frac{1}{4}x = \frac{15}{60}x$
• $-\frac{1}{5}x = -\frac{12}{60}x$
• $\frac{2}{3}x = \frac{40}{60}x$
• $-\frac{2}{5}x = -\frac{24}{60}x$

Step 3: Combine the fractions:

$\frac{15}{60}x - \frac{12}{60}x + \frac{40}{60}x - \frac{24}{60}x$

This simplifies to:

$\frac{15 - 12 + 40 - 24}{60}x = \frac{19}{60}x$

Step 4: Set up the equation:

$\frac{19}{60}x = 1$

Step 5: Solve for $x$ by isolating it on one side of the equation. Multiply both sides by the reciprocal of $\frac{19}{60}$, which is $\frac{60}{19}$:

$x = 1 \times \frac{60}{19}$

$x = \frac{60}{19}$

Therefore, the solution to the problem is $x = \frac{60}{19} = \frac{15}{7}$ after simplifying the fraction.

To solve for $x$ in the equation $\frac{2}{3}x+\frac{1}{4}x-\frac{1}{5}x+\frac{1}{2}x=2$, we will follow these steps:

Step 1: Combine the coefficients of $x$ by finding a common denominator.
- The denominators are 3, 4, 5, and 2. The least common multiple (LCM) of these is 60.
- Rewrite each term with a denominator of 60:
$\frac{2}{3}x = \frac{40}{60}x$, $\frac{1}{4}x = \frac{15}{60}x$, $\frac{1}{5}x = \frac{12}{60}x$, $\frac{1}{2}x = \frac{30}{60}x$.

Step 2: Combine the fractions:
- Combine to get: $\frac{40}{60}x + \frac{15}{60}x - \frac{12}{60}x + \frac{30}{60}x = \frac{73}{60}x.$

Step 3: Set up the equation and solve for $x$:
- The equation becomes $\frac{73}{60}x = 2$.
- Multiply both sides by $\frac{60}{73}$ to isolate $x$:
$x = 2 \times \frac{60}{73}$.

Therefore, $x = \frac{120}{73}$.