Solve each equation using simplification techniques. Round answers to two decimal places as necessary.
$-8 w+3(22+6 w)=17+3 w$
Answer (1)
Distribute the 3: − 8 w + 3 ( 22 + 6 w ) = − 8 w + 66 + 18 w .
Combine like terms: − 8 w + 66 + 18 w = 10 w + 66 .
Isolate w : 10 w + 66 = 17 + 3 w ⟹ 7 w = − 49 .
Solve for w : w = − 7 , which rounded to two decimal places is − 7.00 .
Explanation
Analyze the problem We are given the equation − 8 w + 3 ( 22 + 6 w ) = 17 + 3 w and asked to solve for w . Our goal is to isolate w on one side of the equation by performing algebraic manipulations.
Distribute First, distribute the 3 in the expression 3 ( 22 + 6 w ) : 3 ( 22 + 6 w ) = 3 oc an ce l × 22 + 3 oc an ce l × 6 w = 66 + 18 w
Substitute back into the equation Now, substitute this back into the original equation: − 8 w + 66 + 18 w = 17 + 3 w
Combine like terms Combine like terms on the left side of the equation: − 8 w + 18 w + 66 = 10 w + 66
Rewrite the equation So the equation becomes: 10 w + 66 = 17 + 3 w
Subtract 3w from both sides Subtract 3 w from both sides of the equation: 10 w − 3 w + 66 = 17 + 3 w − 3 w 7 w + 66 = 17
Subtract 66 from both sides Subtract 66 from both sides of the equation: 7 w + 66 − 66 = 17 − 66 7 w = − 49
Divide by 7 Divide both sides by 7 to solve for w :
7 7 w = 7 − 49 w = − 7
Final Answer Since we are asked to round the answer to two decimal places if necessary, we can write − 7 as − 7.00 .
Thus, the solution is w = − 7.00 .
Examples
This type of equation solving is used in many real-world scenarios, such as balancing chemical equations, calculating electrical circuits, or determining the trajectory of a projectile. For example, if you are designing a bridge, you need to calculate the forces acting on the bridge and make sure that the bridge can withstand those forces. This involves solving equations similar to the one above.
