Find the equation of the line through the points (5, 7) parallel to the line 7x + 5y = 12.

1. Analyze the question

We need to find the equation of a line that passes through the point $(5, 7)$ and is parallel to the given line $7x + 5y = 12$.

2. Explain the concept

• Parallel Lines: Two lines are parallel if and only if they have the same slope ($m$).
• Slope-Intercept Form: The equation of a line can be written as $y = mx + c$, where $m$ is the slope.
• General Form: For a line in the form $Ax + By = C$, the slope is given by $m = -\frac{A}{B}$.
• Property: If two lines are parallel, their coefficients of $x$ and $y$ remain the same, but the constant term ($C$) changes. Therefore, a line parallel to $Ax + By = C$ will have the form $Ax + By = k$.

3. Show the solution

Step 1: Identify the slope of the given line. The given line is $7x + 5y = 12$. Using the formula $m = -\frac{A}{B}$, the slope is: $m = -\frac{7}{5}$

Step 2: Use the parallel line property. Since the required line is parallel to $7x + 5y = 12$, it must also have the form: $7x + 5y = k$

Step 3: Solve for $k$ using the given point $(5, 7)$. Substitute $x = 5$ and $y = 7$ into the equation $7x + 5y = k$: $7(5) + 5(7) = k$ $35 + 35 = k$ $70 = k$

Step 4: Write the final equation. Substitute $k = 70$ back into the equation: $7x + 5y = 70$

4. Identify the answer

Comparing our result with the given options, we find that it matches option B.

5. Exam Tip

Whenever you see a question asking for a line parallel to $Ax + By = C$, simply write down $Ax + By = k$ and plug in the coordinates of the given point to find $k$ immediately. This saves you the time of calculating the slope and using the point-slope formula!