Write the standard form of the equation of the line through the given points: (-2, -5) and (-1, 4)
Understand the Problem
The question is asking for the standard form of the equation of a line that passes through two given points, (-2, -5) and (-1, 4). To solve it, we need to calculate the slope using the points, then use the point-slope form to derive the standard form of the equation.
Answer
The standard form of the equation is \( 9x - y = -13 \).
Answer for screen readers
The standard form of the equation of the line is ( 9x - y = -13 ).
Steps to Solve
- Calculate the slope (m)
To find the slope $m$, use the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Using the points $(-2, -5)$ and $(-1, 4)$:
- Let $(x_1, y_1) = (-2, -5)$ and $(x_2, y_2) = (-1, 4)$
- Substitute the values into the formula:
$$ m = \frac{4 - (-5)}{-1 - (-2)} = \frac{4 + 5}{-1 + 2} = \frac{9}{1} = 9 $$
- Use point-slope form of the equation
Now use the point-slope form (y - y_1 = m(x - x_1)) with one of the points. We'll use $(-2, -5)$: $$ y - (-5) = 9(x - (-2)) $$
This simplifies to: $$ y + 5 = 9(x + 2) $$
- Distribute and rearrange to standard form
Distributing gives: $$ y + 5 = 9x + 18 $$
Now, subtract 9x from both sides: $$ -9x + y + 5 = 18 $$
Subtract 5 from both sides: $$ -9x + y = 13 $$
To write it in standard form Ax + By = C, we want A to be positive, so multiply the entire equation by -1: $$ 9x - y = -13 $$
- Write the final standard form
The final standard form of the equation is: $$ 9x - y = -13 $$
The standard form of the equation of the line is ( 9x - y = -13 ).
More Information
The standard form of a linear equation is generally written as ( Ax + By = C ), where A, B, and C are integers. In this case, the slope of the line is 9, indicating a steep incline as the line moves from left to right.
Tips
- Forgetting the signs when calculating the slope, leading to incorrect values.
- Misapplying the point-slope formula can lead to mistakes; make sure to carefully plug in the values.
- Not rearranging the equation to standard form correctly.
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