Which equations have the same value of $x$ as $\frac{5}{6} x+\frac{2}{3}=-9$ ? Select three options.
$6\left(\frac{5}{6} x+\frac{2}{3}\right)=-9$
$6\left(\frac{5}{6} x+\frac{2}{3}\right)=-9(6)$
$5 x+4=-54$
$5 x+4=-9$
$5 x=-13$
$5 x=-58$
Answer (1)
Multiply both sides of the original equation by 6: 6 ( 6 5 x + 3 2 ) = 6 ( − 9 ) which simplifies to 5 x + 4 = − 54 .
Subtract 4 from both sides: 5 x = − 58 .
The three equivalent equations are: 6 ( 6 5 x + 3 2 ) = − 9 ( 6 ) , 5 x + 4 = − 54 , and 5 x = − 58 .
The equations that have the same value of x are 6 ( 6 5 x + 3 2 ) = − 9 ( 6 ) , 5 x + 4 = − 54 , 5 x = − 58 .
Explanation
Analyze the problem We are given the equation 6 5 x + 3 2 = − 9 and asked to find three other equations that have the same solution. Let's start by manipulating the given equation to find equivalent forms.
Multiply by 6 First, multiply both sides of the original equation by 6 to eliminate the fractions: 6 ( 6 5 x + 3 2 ) = 6 ( − 9 )
Distribute Distribute the 6 on the left side: 6 × 6 5 x + 6 × 3 2 = − 54
Simplify Simplify the equation: 5 x + 4 = − 54 This matches one of the options.
Isolate 5x Subtract 4 from both sides of the equation: 5 x + 4 − 4 = − 54 − 4
Simplify Simplify: 5 x = − 58 This matches another one of the options.
Check options Now, let's check the given options to see which ones match the equations we derived.
The first option is 6 ( 6 5 x + 3 2 ) = − 9 . This is not the same as multiplying the right side by 6, so it's not equivalent.
The second option is 6 ( 6 5 x + 3 2 ) = − 9 ( 6 ) . This simplifies to 5 x + 4 = − 54 , which we already found.
The third option is 5 x + 4 = − 54 , which we already found.
The fourth option is 5 x + 4 = − 9 . This is not equivalent to 5 x + 4 = − 54 .
The fifth option is 5 x = − 13 . This is not equivalent to 5 x = − 58 .
The sixth option is 5 x = − 58 , which we already found.
Final Answer Therefore, the three equations that have the same value of x as 6 5 x + 3 2 = − 9 are:
6 ( 6 5 x + 3 2 ) = − 9 ( 6 )
5 x + 4 = − 54
5 x = − 58
Examples
When solving equations in physics or engineering, you often need to manipulate them into different forms to isolate variables or simplify calculations. For example, if you're analyzing a circuit and have an equation relating voltage, current, and resistance, you might multiply both sides by a constant or rearrange terms to solve for a specific variable. Understanding how to create equivalent equations is crucial for problem-solving in these fields.
