Select the correct answer.
Which equation is equivalent to the given equation?
$-4(x-5)+8 x=9 x-3$
A. $5 x=-12$
B. $-5 x=-23$
C. $-5 x=2$
D. $5 x=17$
Answer (1)
Expand the left side of the equation: − 4 ( x − 5 ) + 8 x = 4 x + 20 .
Rewrite the equation: 4 x + 20 = 9 x − 3 .
Isolate x terms: 5 x = 23 .
The equivalent equation is − 5 x = − 23 .
Explanation
Analyze the problem We are given the equation − 4 ( x − 5 ) + 8 x = 9 x − 3 and asked to find an equivalent equation from the options: A. 5 x = − 12 B. − 5 x = − 23 C. − 5 x = 2 D. 5 x = 17 Our goal is to simplify the given equation and see which of the options matches our simplified form.
Expand and simplify the left side First, we expand the left side of the given equation: − 4 ( x − 5 ) + 8 x = − 4 x + 20 + 8 x Combining like terms, we get: − 4 x + 8 x + 20 = 4 x + 20
Rewrite the equation Now, we rewrite the original equation with the simplified left side: 4 x + 20 = 9 x − 3
Subtract 4x from both sides Next, we want to isolate the x terms on one side and the constants on the other side. Let's subtract 4 x from both sides of the equation: 4 x + 20 − 4 x = 9 x − 3 − 4 x 20 = 5 x − 3
Add 3 to both sides Now, add 3 to both sides of the equation: 20 + 3 = 5 x − 3 + 3 23 = 5 x
Rewrite the equation Finally, we can rewrite the equation as: 5 x = 23
Find the correct option Comparing this equation with the given options, we see that none of the options match exactly. However, if we multiply both sides of option B by -1, we get: − 5 x = − 23 ⟹ 5 x = 23 This matches our simplified equation. Therefore, option B is the correct answer.
Final Answer The equation equivalent to − 4 ( x − 5 ) + 8 x = 9 x − 3 is 5 x = 23 , which is the same as − 5 x = − 23 . Therefore, the correct answer is B.
Examples
When solving for an unknown variable in an equation, it's crucial to simplify and isolate the variable. This skill is applicable in various real-life scenarios, such as determining the amount of ingredients needed for a recipe based on the number of servings or calculating the distance traveled based on speed and time. By manipulating equations, we can find the exact values needed to solve practical problems.
