Which equation describes the same line as $y-5=-2(x+4)$?
A. $y=-2 x-2$
B. $y=-2 x-3$
C. $y=-2 x-8$
D. $y=-2 x+9$
Answer (1)
Expand the right side of the given equation: y − 5 = − 2 x − 8 .
Isolate y by adding 5 to both sides: y = − 2 x − 8 + 5 .
Simplify the equation: y = − 2 x − 3 .
The equation that describes the same line is y = − 2 x − 3 .
Explanation
Understanding the Problem We are given the equation y − 5 = − 2 ( x + 4 ) and asked to find an equivalent equation in the form y = m x + b . This involves simplifying the given equation and comparing it to the provided options.
Expanding the Equation First, we expand the right side of the equation:
y − 5 = − 2 ( x + 4 )
y − 5 = − 2 x − 8
Isolating y Next, we isolate y by adding 5 to both sides of the equation:
y − 5 + 5 = − 2 x − 8 + 5
y = − 2 x − 3
Comparing with Options Finally, we compare the simplified equation y = − 2 x − 3 with the given options. Option B, y = − 2 x − 3 , matches our simplified equation.
Final Answer Therefore, the equation that describes the same line as y − 5 = − 2 ( x + 4 ) is y = − 2 x − 3 .
Examples
Understanding linear equations is crucial in many real-world scenarios. For example, if you are saving money, you can model your savings with a linear equation where the slope represents your rate of saving per week and the y-intercept represents your initial savings. Similarly, in physics, the distance traveled at a constant speed can be modeled with a linear equation, where the slope is the speed and the y-intercept is the initial position. Linear equations also appear in economics, such as modeling the cost of production as a function of the number of units produced.
