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 (2)
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 to get x = − 6 ⋅ 3 4 .
The correct one-step operation is 3 4 ( 4 3 ) x = − 6 ( 3 4 ) .
Explanation
Understanding the equation 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 Multiplying both sides of the equation 4 3 x = − 6 by 3 4 , we get 3 4 ⋅ 4 3 x = − 6 ⋅ 3 4 .
Simplifying the equation Simplifying the left side, we have 1 ⋅ x = − 6 ⋅ 3 4 , which simplifies to x = − 6 ⋅ 3 4 .
Identifying the correct option Comparing this with the given options, we see that 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 accidentally added -6 cups. To correct this in one step, you need to multiply the incorrect amount by the reciprocal of 4 3 , which is 3 4 . This example shows how solving equations by multiplying by reciprocals can help correct mistakes in real-life situations, such as adjusting ingredient measurements in cooking or balancing financial accounts.
To solve 4 3 x = − 6 for x in one step, multiply both sides by the reciprocal of 4 3 , which is 3 4 . The correct option is A: 3 4 ( 4 3 ) x = − 6 ( 3 4 ) . This isolates x and leads to the solution.
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