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

Analysis

To find the equation of a line parallel to a given line, we must understand that parallel lines have the same slope (gradient). We are given a point $(5, 7)$ and the equation of a reference line $7x + 5y = 12$.

Concept

1. Slope-Intercept Form: The equation of a line is often written as $y = mx + c$, where $m$ is the slope.
2. Parallel Lines: If two lines are parallel, their slopes are equal ($m_1 = m_2$).
3. Point-Slope Form: Once we have the slope and a point $(x_1, y_1)$, we can use the formula: $y - y_1 = m(x - x_1)$

Step-by-Step Solution

Step 1: Find the slope of the given line. The given equation is $7x + 5y = 12$. Let's rewrite this in the form $y = mx + c$: $5y = -7x + 12$ $y = -\frac{7}{5}x + \frac{12}{5}$ The slope ($m$) of this line is $-\frac{7}{5}$. Since our new line is parallel, its slope must also be $-\frac{7}{5}$.

Step 2: Use the point-slope formula. Using the point $(5, 7)$ and the slope $m = -\frac{7}{5}$: $y - 7 = -\frac{7}{5}(x - 5)$

Step 3: Simplify the equation. Multiply both sides by 5 to clear the fraction: $5(y - 7) = -7(x - 5)$ $5y - 35 = -7x + 35$

Step 4: Rearrange into the standard form ($Ax + By = C$). Add $7x$ to both sides and add $35$ to both sides: $7x + 5y = 35 + 35$ $7x + 5y = 70$

Conclusion

Comparing our result $7x + 5y = 70$ with the given options, it matches option B.

Exam Tip: Whenever you see "parallel to," immediately look at the coefficients of $x$ and $y$ in the original equation. A parallel line will always have the same $Ax + By$ part; you only need to find the new constant ($C$) by plugging in your given coordinates!