Which equations have the same value of [tex]$x$[/tex] as [tex]$\frac{5}{6} x+\frac{2}{3}=-9$[/tex]? Select three options.
[tex]$6\left(\frac{5}{6} x+\frac{2}{3}\right)=-9$[/tex]
[tex]$8\left(\frac{5}{6} x+\frac{2}{3}\right)=-9(6)$[/tex]
[tex]$5 x+4=-54$[/tex]
[tex]$5 x+4=-9$[/tex]
[tex]$5 x=-13$[/tex]
[tex]$5 x=-58$[/tex]
Answer (1)
Solve the equation 6 5 x + 3 2 = − 9 for x , obtaining x = − 5 58 .
Check each of the given equations to see if it has the same solution.
The equations 5 x + 4 = − 54 and 5 x = − 58 have the same solution x = − 5 58 .
The equations with the same value of x are 5 x + 4 = − 54 and 5 x = − 58 .
Explanation
Solve the original equation First, let's solve the given equation for x to find the target value. The equation is:
Equation 6 5 x + 3 2 = − 9
Multiply by 6 To eliminate the fractions, multiply both sides of the equation by 6:
Simplify 6 ( 6 5 x + 3 2 ) = 6 ( − 9 ) 5 x + 4 = − 54
Isolate x Now, isolate x by subtracting 4 from both sides:
Subtract 4 5 x = − 54 − 4 5 x = − 58
Solve for x Finally, divide both sides by 5 to solve for x :
Divide by 5 x = 5 − 58
Check each equation Now we need to check each of the provided equations to see which ones have the same solution, x = − 5 58 .
Equation 1 Equation 1: 6 ( 6 5 x + 3 2 ) = − 9 . This simplifies to 5 x + 4 = − 9 . Solving for x , we get 5 x = − 13 , so x = − 5 13 . This is not equal to − 5 58 .
Equation 2 Equation 2: 8 ( 6 5 x + 3 2 ) = − 9 ( 6 ) . This simplifies to 6 40 x + 3 16 = − 54 . Multiplying by 3, we get 3 20 x + 16 = − 162 . Then 3 20 x = − 178 , so x = − 178 ⋅ 20 3 = − 10 89 ⋅ 3 = − 10 267 . This is not equal to − 5 58 .
Equation 3 Equation 3: 5 x + 4 = − 54 . Solving for x , we get 5 x = − 58 , so x = − 5 58 . This matches our target value.
Equation 4 Equation 4: 5 x + 4 = − 9 . Solving for x , we get 5 x = − 13 , so x = − 5 13 . This is not equal to − 5 58 .
Equation 5 Equation 5: 5 x = − 13 . Solving for x , we get x = − 5 13 . This is not equal to − 5 58 .
Equation 6 Equation 6: 5 x = − 58 . Solving for x , we get x = − 5 58 . This matches our target value.
Conclusion Therefore, the equations that have the same value of x as the original equation are:
Final Answer 5 x + 4 = − 54 and 5 x = − 58 . Also, from our steps in solving the original equation, we found 5 x + 4 = − 54 .
Examples
Imagine you're baking a cake and need to adjust a recipe. The original recipe calls for 6 5 cups of flour, but you want to use a different amount to make the cake sweeter. If you know the desired sweetness corresponds to a total 'sweetness level' of -9, and you also know that adding 3 2 cups of sugar will affect the sweetness, you can use the equation 6 5 x + 3 2 = − 9 to find the exact amount of flour ( x ) needed to achieve the desired sweetness. This type of problem helps in adjusting proportions in recipes or any situation where you need to balance different components to achieve a specific outcome.
