Write the equation of a line that passes through the points shown in the table.
x
-10
-4
8
14
Which equations represent a line that passes through the points given in the table? Check all that apply.
y-2=-6(x+10)
y-2=-\frac{1}{6}(x+10)
y-1=-\frac{1}{6}(x+4)
y=-6 x-58
y=-\frac{1}{6} x+\frac{1}{3}
y=-\frac{1}{6} x+5
Answer (2)
The equations of the line that passes through the points assumed from the table are: y - 2 = -\frac{1}{6}(x + 10), y - 1 = -\frac{1}{6}(x + 4), and y = -\frac{1}{6}x + \frac{1}{3}.
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Calculate the slope using two points from the table: m = x 2 − x 1 y 2 − y 1 = − 6 1 .
Use the point-slope form with one point and the slope: y − 2 = − 6 1 ( x + 10 ) .
Convert to slope-intercept form: y = − 6 1 x + 3 1 .
The equations that represent the line are: y − 2 = − 6 1 ( x + 10 ) , y − 1 = − 6 1 ( x + 4 ) , and y = − 6 1 x + 3 1 .
y − 2 = − 6 1 ( x + 10 ) , y − 1 = − 6 1 ( x + 4 ) , y = − 6 1 x + 3 1
Explanation
Problem Analysis First, let's analyze the problem. We are given a table with x-values and asked to find the equation of a line that passes through the points defined by these x-values. We are also given a set of equations and need to determine which of them represent the same line. To do this, we need to find the y-values corresponding to the given x-values. Since we are not given the y-values, we will assume that the line passes through the points (-10, 2) and (-4, 1).
Calculating the Slope Next, we calculate the slope (m) of the line using the points (-10, 2) and (-4, 1). The formula for the slope is: m = x 2 − x 1 y 2 − y 1 Substituting the given points: m = − 4 − ( − 10 ) 1 − 2 = 6 − 1 = − 6 1
Point-Slope Form Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is: y − y 1 = m ( x − x 1 ) Using the point (-10, 2) and the slope m = − 6 1 :
y − 2 = − 6 1 ( x − ( − 10 )) y − 2 = − 6 1 ( x + 10 )
Slope-Intercept Form Let's convert the point-slope form to slope-intercept form ( y = m x + b ) to make it easier to compare with the given equations: y − 2 = − 6 1 x − 6 10 y = − 6 1 x − 3 5 + 2 y = − 6 1 x − 3 5 + 3 6 y = − 6 1 x + 3 1
Checking the Equations Now, we check each of the given equations to see if they match the equations we derived:
y − 2 = − 6 ( x + 10 ) : This equation has a slope of -6, which does not match our calculated slope of − 6 1 .
y − 2 = − 6 1 ( x + 10 ) : This equation matches the point-slope form we derived.
y − 1 = − 6 1 ( x + 4 ) : Let's check if the point (-4, 1) satisfies the equation y = − 6 1 x + 3 1 . Substituting x = -4, we get y = − 6 1 ( − 4 ) + 3 1 = 3 2 + 3 1 = 1 . So, this equation is also correct.
y = − 6 x − 58 : This equation has a slope of -6, which does not match our calculated slope of − 6 1 .
y = − 6 1 x + 3 1 : This equation matches the slope-intercept form we derived.
y = − 6 1 x + 5 : This equation has the correct slope but a different y-intercept.
Final Answer Therefore, the equations that represent the line passing through the points in the table are:
y − 2 = − 6 1 ( x + 10 ) y − 1 = − 6 1 ( x + 4 ) y = − 6 1 x + 3 1
Examples
Understanding linear equations is crucial in many real-world applications. For instance, in economics, you can model the relationship between the price of a product and the quantity demanded using a linear equation. If you know two price-quantity points, you can determine the demand curve equation and predict how demand will change with price variations. Similarly, in physics, the relationship between distance, time, and constant velocity is linear, allowing you to predict the position of an object at any given time. Linear equations are also fundamental in computer graphics for rendering lines and shapes.
