Find the parallel line of y=-x+2 that passes through the point (3, 1).

Understand the Problem

The question is asking for the equation of a line that is parallel to the given line, y = -x + 2, and also passes through a specific point, (3, 1). The approach to solve it involves identifying the slope of the given line and using that slope with the point-slope form of a linear equation to find the required line.

Answer

The equation of the line is $ y = -x + 4 $.

Steps to Solve

Identify the slope of the given line

For the line given by the equation $y = -x + 2$, the slope (m) is the coefficient of $x$. Thus, $$ m = -1 $$

Use the point-slope form of the equation

The point-slope form of a line is given by the formula: $$ y - y_1 = m(x - x_1) $$ Here, $(x_1, y_1)$ is the point through which the line passes, which is (3, 1) in this case. Substitute $m = -1$, $x_1 = 3$, and $y_1 = 1$ into the equation:

$$ y - 1 = -1(x - 3) $$

Simplify the equation

Distributing the slope on the right side: $$ y - 1 = -1x + 3 $$

Now, add 1 to both sides to solve for $y$:

$$ y = -x + 4 $$

Thus, the equation of the line parallel to the given line and passing through the point (3, 1) is $$ y = -x + 4 $$

The equation of the line is ( y = -x + 4 ).

More Information

This line is parallel to the original line ( y = -x + 2 ) because they have the same slope of -1. Parallel lines never intersect, maintaining the same steepness.

Tips

Some common mistakes include:

Forgetting that parallel lines have the same slope.
Misapplying the point-slope form of the linear equation.

To avoid these mistakes, always double-check that you correctly identify the slope and correctly substitute the point into the point-slope form.

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