Find the equation of a line in slope intercept form with points $(9,-2)$ and $(-3,2)$.
a. $y=-3 x+2$
b. $y=\frac{-1}{3} x+1$
c. $y=9 x-2$
d. $y=\frac{3}{4} x+5$
Answer (2)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 3 1 .
Use the point-slope form of a line y − y 1 = m ( x − x 1 ) with point ( 9 , − 2 ) to get y + 2 = − 3 1 ( x − 9 ) .
Convert to slope-intercept form by solving for y , resulting in y = − 3 1 x + 1 .
The equation of the line is y = − 3 1 x + 1 .
Explanation
Problem Analysis We are given two points, ( 9 , − 2 ) and ( − 3 , 2 ) , and we want to find the equation of the line that passes through these points in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Calculating the Slope First, we need to find the slope of the line. The slope, m , is given by the formula: m = x 2 − x 1 y 2 − y 1 Using the given points ( 9 , − 2 ) and ( − 3 , 2 ) , we have: m = − 3 − 9 2 − ( − 2 ) = − 12 4 = − 3 1 So, the slope of the line is − 3 1 .
Using Point-Slope Form Now that we have the slope, we can use the point-slope form of a line, which is: y − y 1 = m ( x − x 1 ) We can use either of the given points. Let's use ( 9 , − 2 ) . Plugging in the values, we get: y − ( − 2 ) = − 3 1 ( x − 9 ) y + 2 = − 3 1 x + 3
Converting to Slope-Intercept Form Now, we convert the equation to slope-intercept form by solving for y :
y = − 3 1 x + 3 − 2 y = − 3 1 x + 1 So, the equation of the line in slope-intercept form is y = − 3 1 x + 1 .
Final Answer The equation of the line in slope-intercept form is y = − 3 1 x + 1 . Comparing this to the given options, we see that option b matches our result.
Examples
Understanding linear equations is crucial in many real-world applications. For instance, in economics, you might use a linear equation to model the relationship between the price of a product and the quantity demanded. If you know two price-quantity data points, you can determine the demand curve equation, helping you predict how changes in price will affect demand. Similarly, in physics, you can model the motion of an object moving at a constant velocity using a linear equation, where the slope represents the velocity and the y-intercept represents the initial position.
The equation of the line in slope-intercept form passing through the points (9, -2) and (-3, 2) is y = − 3 1 x + 1 . This corresponds to option (b) from the provided choices.
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