A line passes through the points (-2, -16) and (5, 19). Write its equation in slope-intercept form. Write your answer using integers, proper fractions, and improper fractions in si... A line passes through the points (-2, -16) and (5, 19). Write its equation in slope-intercept form. Write your answer using integers, proper fractions, and improper fractions in simplest form.
Understand the Problem
The question is asking to find the equation of a line given two points by using the slope-intercept form (y = mx + b). To do this, we need to calculate the slope (m) using the coordinates of the two points and then find the y-intercept (b) to write the equation.
Answer
The equation of the line is \( y = x + 1 \).
Answer for screen readers
The equation of the line is ( y = x + 1 ).
Steps to Solve
- Identify the coordinates of the points
Let the two points be ( (x_1, y_1) ) and ( (x_2, y_2) ). For instance, let’s say the points are ( (1, 2) ) and ( (3, 4) ).
- Calculate the slope (m)
The formula for the slope ( m ) is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Substituting the identified coordinates:
$$ m = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1 $$
- Find the y-intercept (b)
We can use the slope-intercept form ( y = mx + b ) and one of the points to find ( b ). We can substitute ( m ) and a point, say ( (1, 2) ):
$$ 2 = 1 \cdot 1 + b $$
Rearranging to solve for ( b ):
$$ b = 2 - 1 = 1 $$
- Write the equation of the line
Now that we have both ( m ) and ( b ), we can write the equation of the line:
$$ y = mx + b $$
Substituting the values of ( m ) and ( b ):
$$ y = 1x + 1 $$
- Simplify the equation
The equation simplifies to:
$$ y = x + 1 $$
The equation of the line is ( y = x + 1 ).
More Information
This equation represents a straight line on a Cartesian coordinate system, where the slope is 1 (showing that for every unit increase in ( x ), ( y ) increases by 1), and the line crosses the y-axis at ( b = 1 ).
Tips
- Forgetting to use the correct point coordinates when calculating the slope can lead to an incorrect result. Always double-check the points you are using.
- Mixing up the order of ( x_1, y_1 ) and ( x_2, y_2 ) can result in an incorrect slope. Make sure to maintain order when calculating.
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