Use properties to rewrite the given equation. Which equations have the same solution as the equation
[tex]\frac{3}{5} x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5} x[/tex]
Select three options.
A. [tex]\frac{8}{5} x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5} x[/tex]
B. [tex]18 x+20+30 x=15-6 x[/tex]
C. [tex]18 x+20+x=15-6 x[/tex]
D. [tex]24 x+30 x=-5[/tex]
E. [tex]12 x+30 x=-5[/tex]
Answer (1)
Combine like terms: Rewrite the original equation by combining like terms on the left side, resulting in 5 8 x + 3 2 = 2 1 − 5 1 x .
Eliminate fractions: Multiply the original equation by 30 to eliminate fractions, leading to 18 x + 20 + 30 x = 15 − 6 x .
Simplify and rearrange: Combine terms and rearrange the equation to isolate x, resulting in 24 x + 30 x = − 5 .
The three equivalent equations are: 5 8 x + 3 2 = 2 1 − 5 1 x , 18 x + 20 + 30 x = 15 − 6 x , 24 x + 30 x = − 5
Explanation
Analyzing the Problem We are given the equation 5 3 x + 3 2 + x = 2 1 − 5 1 x and asked to find three equivalent equations from the given options.
Simplifying the Left Side First, let's simplify the left side of the equation by combining the x terms: 5 3 x + x = 5 3 x + 5 5 x = 5 8 x . So the equation becomes 5 8 x + 3 2 = 2 1 − 5 1 x . This matches the first option.
Eliminating Fractions Next, let's eliminate the fractions in the original equation by multiplying both sides by the least common multiple (LCM) of the denominators (5, 3, 2), which is 30. This gives 30 ( 5 3 x + 3 2 + x ) = 30 ( 2 1 − 5 1 x ) .
Distributing Distribute the 30 on both sides: 30 ( 5 3 x ) + 30 ( 3 2 ) + 30 ( x ) = 30 ( 2 1 ) − 30 ( 5 1 x ) . This simplifies to 18 x + 20 + 30 x = 15 − 6 x . This matches the second option.
Checking Option 3 Now, let's examine the remaining options. Option 3 is 18 x + 20 + x = 15 − 6 x , which is not equivalent to the original equation or the equation we derived in step 4.
Checking Option 4 Option 4 is 24 x + 30 x = − 5 . Let's start from the equation we found in step 4: 18 x + 20 + 30 x = 15 − 6 x . Combining like terms, we get 48 x + 20 = 15 − 6 x . Adding 6 x to both sides and subtracting 20 from both sides, we get 54 x = − 5 . Option 4 can be written as 54 x = − 5 , so it is equivalent to the original equation.
Checking Option 5 Option 5 is 12 x + 30 x = − 5 , which simplifies to 42 x = − 5 . This is not equivalent to 54 x = − 5 , so it is not a correct option.
Final Answer Therefore, the three equations that have the same solution as the given equation are: 5 8 x + 3 2 = 2 1 − 5 1 x 18 x + 20 + 30 x = 15 − 6 x 24 x + 30 x = − 5
Examples
When solving equations, it's helpful to find equivalent forms that are easier to work with. For instance, if you're calculating the dimensions of a rectangular garden with a fixed area and perimeter, you might start with a complex equation relating length and width. By finding equivalent equations through algebraic manipulation, you can simplify the problem and more easily determine the possible dimensions of the garden. This skill is also useful in physics, where you might need to rearrange equations to solve for a specific variable, such as velocity or acceleration.
