Which shows how to solve the equation [tex]$\frac{3}{4} x=-6$[/tex] for [tex]$x$[/tex] in one step?
A. [tex]$\frac{4}{3}\left(\frac{3}{4}\right) x=-6\left(\frac{4}{3}\right)$[/tex]
B. [tex]$4\left(\frac{3}{4}\right) x=-6(4)$[/tex]
C. [tex]$4\left(\frac{3}{4}\right) x=-6(3)$[/tex]
D. [tex]$\frac{4}{3}\left(\frac{3}{4}\right) x=-6\left(\frac{3}{4}\right)$[/tex]
Answer (1)
To solve the equation 4 3 x = − 6 for x in one step:
Multiply both sides of the equation by the reciprocal of 4 3 , which is 3 4 .
This gives 3 4 ⋅ 4 3 x = − 6 ⋅ 3 4 .
Simplify the left side to get x .
Simplify the right side to get x = − 8 .
The correct one-step operation is 3 4 ( 4 3 ) x = − 6 ( 3 4 ) .
Explanation
Understanding the Problem We are given the equation 4 3 x = − 6 and asked to identify the correct one-step operation to solve for x . To isolate x , we need to multiply both sides of the equation by the reciprocal of the coefficient of x , which is 4 3 . The reciprocal of 4 3 is 3 4 .
Isolating x Multiply both sides of the equation by 3 4 : 3 4 ⋅ 4 3 x = − 6 ⋅ 3 4
Simplifying the Equation Simplify the left side: 3 4 ⋅ 4 3 x = 1 ⋅ x = x Simplify the right side: − 6 ⋅ 3 4 = 3 − 6 ⋅ 4 = 3 − 24 = − 8 So, we have x = − 8 .
Identifying the Correct Option Comparing the result with the given options, the correct one-step operation is: 3 4 ( 4 3 ) x = − 6 ( 3 4 )
Examples
Imagine you're baking a cake and the recipe calls for 4 3 cup of flour, but you only want to make a smaller cake that requires -6 cups equivalent. To find out how much of the smaller cake you can make, you need to solve the equation 4 3 x = − 6 . Multiplying both sides by 3 4 helps you determine the scaling factor x , which tells you how much to adjust all the ingredients. This principle applies to various scaling problems in cooking, construction, and other fields.
