Which of the following is the standard form of a line with slope of 2 passing through the point (-1, 3)? A) 2x - y = -5 B) 2x + y = 5 C) 2x + y = -5 D) 2x - y = 5
Understand the Problem
The question is asking for the standard form of a line given a slope and a specific point. To solve it, we can use the slope-intercept form to derive the standard form equation of the line. First, we will write the equation in point-slope form using the given slope and point, then convert it to standard form.
Answer
The standard form of the line is given by: $$ mx - y = ma - b $$
Answer for screen readers
The standard form of the line is given by the equation:
$$ mx - y = ma - b $$
Steps to Solve
- Write the point-slope form of the equation
Using the slope ($m$) and the point $(x_1, y_1)$, the point-slope form of the equation is:
$$ y - y_1 = m(x - x_1) $$
- Substitute the values into the point-slope form
Assuming the given slope is $m$ and the point is $(a, b)$, substitute these values into the equation:
$$ y - b = m(x - a) $$
- Distribute the slope
Distribute $m$ to both $(x - a)$:
$$ y - b = mx - ma $$
- Rearrange to get the slope-intercept form
To isolate $y$, add $b$ to both sides:
$$ y = mx + (b - ma) $$
- Convert to standard form
Standard form is $Ax + By = C$. To convert, rearrange the equation:
- Move $mx$ to the left side:
$$ -mx + y = (b - ma) $$
- Multiply through by -1 to have $A$ as a positive coefficient:
$$ mx - y = ma - b $$
Now, we can express this in standard form as:
$$ Ax + By = C $$
where $A = m$, $B = -1$, and $C = ma - b$.
The standard form of the line is given by the equation:
$$ mx - y = ma - b $$
More Information
The standard form of a line is useful because it allows quick identification of the coefficients, which represent the direction and position of the line in the Cartesian plane. It is especially handy for solving systems of equations.
Tips
- Forgetting to distribute the slope correctly when converting from point-slope to slope-intercept form.
- Not rearranging to get $Ax + By = C$ format properly.
- Confusing standard form with slope-intercept form.
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