Find an equation of the line that has slope 2 and passes through (1, -3).
Understand the Problem
The question asks for the equation of a line that has a specific slope and passes through a given point. To solve this, we will use the point-slope form of a linear equation.
Answer
The equation of the line is $2x - y - 5 = 0$.
Answer for screen readers
The equation of the line is:
$$ 2x - y - 5 = 0 $$
Steps to Solve
- Identify the point-slope form of the equation
The point-slope form of a linear equation is given by the formula:
$$ y - y_1 = m(x - x_1) $$
where ( m ) is the slope, and ( (x_1, y_1) ) is a point on the line.
- Plug in the values
We have ( m = 2 ) and the point ( (x_1, y_1) = (1, -3) ). Substituting these values into the equation yields:
$$ y - (-3) = 2(x - 1) $$
- Simplify the equation
First, simplify ( y - (-3) ) to ( y + 3 ):
$$ y + 3 = 2(x - 1) $$
Then expand the right side:
$$ y + 3 = 2x - 2 $$
- Isolate ( y )
To isolate ( y ), subtract 3 from both sides:
$$ y = 2x - 2 - 3 $$
This simplifies to:
$$ y = 2x - 5 $$
- Convert to standard form (optional)
To convert the equation into standard form, rearrange it to:
$$ 2x - y - 5 = 0 $$
The equation of the line is:
$$ 2x - y - 5 = 0 $$
More Information
This equation represents a linear relationship where the slope is 2, indicating the line rises 2 units for every 1 unit it moves to the right. The point (1, -3) lies on the line.
Tips
- Forgetting to represent the slope as a positive or negative number: Ensure you identify the slope correctly whenever substituting.
- Misplacing the coordinates: Always double-check that you are substituting the correct values for ( x_1 ) and ( y_1 ).
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