Answer :
To solve the given linear equation using the distributive property, let’s break it down step by step.
Given equation:
[tex]\[ \frac{3}{5}(x - 10) = 18 - 4x - 1 \][/tex]
### Step 1: Applying the Distributive Property
The distributive property allows us to multiply a single term by each term inside the parentheses. Here, the fraction \(\frac{3}{5}\) will be distributed across the terms inside the parentheses, \(x\) and \(-10\).
So, we need to distribute the number:
[tex]\[ \frac{3}{5} \][/tex]
### Step 2: Distribute \(\frac{3}{5}\) across \(x - 10\)
[tex]\[ \frac{3}{5}(x - 10) = \left( \frac{3}{5} \right) x - \left( \frac{3}{5} \right) \cdot 10 \][/tex]
This gives:
[tex]\[ \frac{3}{5} x - \frac{30}{5} \][/tex]
Simplify \(\frac{30}{5}\):
[tex]\[ \frac{3}{5} x - 6 \][/tex]
So, the left side of the equation becomes:
[tex]\[ \frac{3}{5} x - 6 \][/tex]
Now, the equation looks like this:
[tex]\[ \frac{3}{5} x - 6 = 18 - 4x - 1 \][/tex]
### Step 3: Simplify the Right-hand Side
Combine like terms on the right-hand side:
[tex]\[ 18 - 1 - 4x = 17 - 4x \][/tex]
So, the equation now is:
[tex]\[ \frac{3}{5} x - 6 = 17 - 4x \][/tex]
### Step 4: Isolate \(x\)
To isolate \(x\), we need to get all \(x\)-terms on one side and the constant terms on the other side. First, let's add \(4x\) to both sides to eliminate the \(-4x\) term on the right-hand side:
[tex]\[ \frac{3}{5} x + 4x - 6 = 17 \][/tex]
Convert \(4x\) to a fraction with a denominator of \(5\) to combine with \(\frac{3}{5} x\):
[tex]\[ \frac{3}{5} x + \frac{20}{5} x - 6 = 17 \][/tex]
Combine the \(x\)-terms:
[tex]\[ \frac{23}{5} x - 6 = 17 \][/tex]
Next, add 6 to both sides to isolate the term with \(x\):
[tex]\[ \frac{23}{5} x = 23 \][/tex]
### Step 5: Solve for \(x\)
Multiply both sides by \(\frac{5}{23}\) to solve for \(x\):
[tex]\[ x = 23 \cdot \frac{5}{23} \][/tex]
This simplifies to:
[tex]\[ x = 5 \][/tex]
Therefore, the solution to the equation is:
[tex]\[ x = 5 \][/tex]
Given equation:
[tex]\[ \frac{3}{5}(x - 10) = 18 - 4x - 1 \][/tex]
### Step 1: Applying the Distributive Property
The distributive property allows us to multiply a single term by each term inside the parentheses. Here, the fraction \(\frac{3}{5}\) will be distributed across the terms inside the parentheses, \(x\) and \(-10\).
So, we need to distribute the number:
[tex]\[ \frac{3}{5} \][/tex]
### Step 2: Distribute \(\frac{3}{5}\) across \(x - 10\)
[tex]\[ \frac{3}{5}(x - 10) = \left( \frac{3}{5} \right) x - \left( \frac{3}{5} \right) \cdot 10 \][/tex]
This gives:
[tex]\[ \frac{3}{5} x - \frac{30}{5} \][/tex]
Simplify \(\frac{30}{5}\):
[tex]\[ \frac{3}{5} x - 6 \][/tex]
So, the left side of the equation becomes:
[tex]\[ \frac{3}{5} x - 6 \][/tex]
Now, the equation looks like this:
[tex]\[ \frac{3}{5} x - 6 = 18 - 4x - 1 \][/tex]
### Step 3: Simplify the Right-hand Side
Combine like terms on the right-hand side:
[tex]\[ 18 - 1 - 4x = 17 - 4x \][/tex]
So, the equation now is:
[tex]\[ \frac{3}{5} x - 6 = 17 - 4x \][/tex]
### Step 4: Isolate \(x\)
To isolate \(x\), we need to get all \(x\)-terms on one side and the constant terms on the other side. First, let's add \(4x\) to both sides to eliminate the \(-4x\) term on the right-hand side:
[tex]\[ \frac{3}{5} x + 4x - 6 = 17 \][/tex]
Convert \(4x\) to a fraction with a denominator of \(5\) to combine with \(\frac{3}{5} x\):
[tex]\[ \frac{3}{5} x + \frac{20}{5} x - 6 = 17 \][/tex]
Combine the \(x\)-terms:
[tex]\[ \frac{23}{5} x - 6 = 17 \][/tex]
Next, add 6 to both sides to isolate the term with \(x\):
[tex]\[ \frac{23}{5} x = 23 \][/tex]
### Step 5: Solve for \(x\)
Multiply both sides by \(\frac{5}{23}\) to solve for \(x\):
[tex]\[ x = 23 \cdot \frac{5}{23} \][/tex]
This simplifies to:
[tex]\[ x = 5 \][/tex]
Therefore, the solution to the equation is:
[tex]\[ x = 5 \][/tex]
