A line is drawn through $(-7,11)$ and $(8,-9)$. The equation $y-11=\frac{-4}{3}(x+7)$ is written to represent the line. Which equations also represent the line? Check all that apply.
$y=\frac{-4}{3} x+\frac{5}{3}$
$3 y=-4 x+40$
$4 x+y^{-5} 21$
$4 x+3 y=5$
$-4 x+3 y=17$
Answer (2)
The equations that represent the same line as the given equation are y = 3 − 4 x + 3 5 and 4 x + 3 y = 5 .
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The problem provides a line equation in point-slope form and asks to identify equivalent equations.
Convert the point-slope form to slope-intercept form: y = 3 − 4 x + 3 5 .
Transform each given equation into slope-intercept form and compare it with the derived equation.
The equations y = 3 − 4 x + 3 5 and 4 x + 3 y = 5 also represent the same line. y = 3 − 4 x + 3 5 , 4 x + 3 y = 5
Explanation
Understanding the Problem We are given a line that passes through the points ( − 7 , 11 ) and ( 8 , − 9 ) . The equation of the line is given as y − 11 = 3 − 4 ( x + 7 ) . We need to determine which of the given equations also represent the same line.
Verifying the Given Equation First, let's verify that the given equation y − 11 = 3 − 4 ( x + 7 ) represents the line passing through ( − 7 , 11 ) and ( 8 , − 9 ) . The slope of the line passing through these two points is m = 8 − ( − 7 ) − 9 − 11 = 15 − 20 = 3 − 4 . The given equation is in point-slope form, using the point ( − 7 , 11 ) and the slope 3 − 4 , so it is correct.
Converting to Slope-Intercept Form Now, let's transform the given equation into slope-intercept form ( y = m x + b ).
y − 11 = 3 − 4 ( x + 7 )
y − 11 = 3 − 4 x − 3 28
y = 3 − 4 x − 3 28 + 11
y = 3 − 4 x − 3 28 + 3 33
y = 3 − 4 x + 3 5
Comparing with Given Equations Now we compare this slope-intercept form with the given options:
y = 3 − 4 x + 3 5 : This is the same as our slope-intercept form, so it represents the same line.
3 y = − 4 x + 40 : Let's convert this to slope-intercept form by dividing by 3: y = 3 − 4 x + 3 40 . This is different from our slope-intercept form, so it does not represent the same line.
4 x + y = 21 : Let's convert this to slope-intercept form: y = − 4 x + 21 . This is different from our slope-intercept form, so it does not represent the same line.
4 x + 3 y = 5 : Let's convert this to slope-intercept form: 3 y = − 4 x + 5 , so y = 3 − 4 x + 3 5 . This is the same as our slope-intercept form, so it represents the same line.
− 4 x + 3 y = 17 : Let's convert this to slope-intercept form: 3 y = 4 x + 17 , so y = 3 4 x + 3 17 . This is different from our slope-intercept form, so it does not represent the same line.
Final Answer Therefore, the equations that also represent the line are:
y = 3 − 4 x + 3 5 and 4 x + 3 y = 5 .
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. Similarly, in physics, the relationship between distance, speed, and time can be represented linearly when the speed is constant. By manipulating and understanding these equations, one can make predictions and informed decisions in these fields.
