Let's solve the problem step by step:
### Step 1: Original Linear Equation
The original linear equation given is:
[tex]\[ 4x + 7y = 13 \][/tex]
### Step 2: Product of 2 and the Linear Equation
To find the product of 2 and the original equation, we multiply each term by 2:
[tex]\[ 2 \cdot (4x + 7y) = 2 \cdot 13 \][/tex]
This gives us:
[tex]\[ 8x + 14y = 26 \][/tex]
So, the product of 2 and the linear equation \(4x + 7y = 13\) is:
[tex]\[ 8x + 14y - 26 \][/tex]
### Step 3: Solving the Original Equation for \(y\)
To express the original equation in slope-intercept form (\(y = mx + b\)), we solve for \(y\):
[tex]\[ 4x + 7y = 13 \][/tex]
First, isolate the term containing \(y\):
[tex]\[ 7y = 13 - 4x \][/tex]
Next, divide by 7:
[tex]\[ y = \frac{13}{7} - \frac{4x}{7} \][/tex]
So, the original equation in slope-intercept form is:
[tex]\[ y = \frac{13}{7} - \frac{4x}{7} \][/tex]
### Step 4: Solving the New Equation for \(y\)
Now we solve the new equation \(8x + 14y = 26\) for \(y\):
[tex]\[ 8x + 14y = 26 \][/tex]
First, isolate the term containing \(y\):
[tex]\[ 14y = 26 - 8x \][/tex]
Next, divide by 14:
[tex]\[ y = \frac{26}{14} - \frac{8x}{14} \][/tex]
Simplify the fractions:
[tex]\[ y = \frac{13}{7} - \frac{4x}{7} \][/tex]
So, the new equation in slope-intercept form is:
[tex]\[ y = \frac{13}{7} - \frac{4x}{7} \][/tex]
### Summary
- The product of 2 and the original equation is \(8x + 14y - 26\)
- The original equation in slope-intercept form is \( y = \frac{13}{7} - \frac{4x}{7} \)
- The new equation in slope-intercept form is [tex]\( y = \frac{13}{7} - \frac{4x}{7} \)[/tex]
