Which equation describes the line that passes through the points $(1,3)$ and $(-2,6)$?
A. $y=-2 x+3$
B. $y=-2 x+6$
C. $y=-x+3$
D. $y=-x+4$
Answer (1)
• Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 : m = − 2 − 1 6 − 3 = − 1 .
• Use the point-slope form of a line: y − y 1 = m ( x − x 1 ) , which gives y − 3 = − 1 ( x − 1 ) .
• Simplify the equation to slope-intercept form: y = − x + 1 + 3 , which simplifies to y = − x + 4 .
• The equation of the line is y = − x + 4 .
Explanation
Understanding the Problem We are given two points, ( 1 , 3 ) and ( − 2 , 6 ) , and we want to find the equation of the line that passes through these points.
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 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points. In our case, ( x 1 , y 1 ) = ( 1 , 3 ) and ( x 2 , y 2 ) = ( − 2 , 6 ) . Plugging these values into the formula, we get: m = − 2 − 1 6 − 3 = − 3 3 = − 1
Using the Point-Slope Form Now that we have the slope, we can use the point-slope form of a line, which is given by: y − y 1 = m ( x − x 1 ) where m is the slope and ( x 1 , y 1 ) is a point on the line. We can use either of the given points; let's use ( 1 , 3 ) . Plugging in the values, we get: y − 3 = − 1 ( x − 1 ) y − 3 = − x + 1
Simplifying to Slope-Intercept Form Now, we can solve for y to get the equation in slope-intercept form ( y = m x + b ): y = − x + 1 + 3 y = − x + 4
Finding the Correct Equation The equation of the line is y = − x + 4 . Comparing this to the given options, we see that it matches the fourth option.
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you're tracking the depreciation of a car's value over time, you might find that it loses a fixed amount each year. This situation can be modeled using a linear equation, where the car's value is the dependent variable (y), the number of years is the independent variable (x), and the rate of depreciation is the slope (m). By knowing the initial value and the rate of depreciation, you can predict the car's value at any point in the future. Similarly, linear equations are used in physics to describe motion with constant velocity, in economics to model supply and demand curves, and in everyday life to calculate costs based on a fixed rate.
