A line that passes through the points $(-4,10)$ and $(-1,5)$ can be represented by the equation $y=-\frac{5}{3}(x-2)$. Which equations also represent this line? Select three options.
$y=-\frac{5}{3} x-2$
$y=-\frac{5}{3} x+\frac{10}{3}$
$3 y=-5 x+10$
$3 x+15 y=30$
$5 x+3 y=10$
Answer (2)
Distribute − 3 5 in the original equation: y = − 3 5 ( x − 2 ) becomes y = − 3 5 x + 3 10 .
Check each option for equivalence to the simplified equation.
Option 2, 3, and 5 are equivalent to the simplified equation.
The three equivalent equations are: y = − 3 5 x + 3 10 , 3 y = − 5 x + 10 , and 5 x + 3 y = 10 .
Therefore, the final answer is y = − 3 5 x + 3 10 , 3 y = − 5 x + 10 , 5 x + 3 y = 10
Explanation
Analyze the problem We are given the equation of a line y = − 3 5 ( x − 2 ) and need to find three equivalent equations from the given options. Let's start by simplifying the given equation.
Simplify the given equation First, distribute the − 3 5 to both terms inside the parenthesis: y = − 3 5 x + 3 10 Now, we will check each of the given options to see if they are equivalent to this equation.
Check Option 1 Option 1: y = − 3 5 x − 2 . This is not equivalent to y = − 3 5 x + 3 10 because the constant term is different.
Check Option 2 Option 2: y = − 3 5 x + 3 10 . This is exactly the same as our simplified equation, so it is equivalent.
Check Option 3 Option 3: 3 y = − 5 x + 10 . Let's divide both sides by 3 to isolate y :
y = − 3 5 x + 3 10 This is the same as our simplified equation, so it is equivalent.
Check Option 4 Option 4: 3 x + 15 y = 30 . Let's isolate y :
15 y = − 3 x + 30 y = − 15 3 x + 15 30 y = − 5 1 x + 2 This is not equivalent to y = − 3 5 x + 3 10 , so it is not a correct option.
Check Option 5 Option 5: 5 x + 3 y = 10 . Let's isolate y :
3 y = − 5 x + 10 y = − 3 5 x + 3 10 This is the same as our simplified equation, so it is equivalent.
Final Answer The three equations that also represent the given line are: y = − 3 5 x + 3 10 3 y = − 5 x + 10 5 x + 3 y = 10
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 points on this demand curve, you can determine the equation of the line and make predictions about how changes in price will affect demand. Similarly, in physics, you can use linear equations to describe the motion of an object at a constant velocity. Knowing two points of the object's position over time allows you to find the equation and predict its future location.
The equivalent equations for the line given by y = − 3 5 ( x − 2 ) are y = − 3 5 x + 3 10 , 3 y = − 5 x + 10 , and 5 x + 3 y = 10 . These options maintain the same slope and intercept after simplification. Hence, they accurately represent the same linear relationship.
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