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 \( y = -x + 4 \).
Answer for screen readers
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.