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Chapter 3: Problem 55

Solve the equation. $$6(u-1)+5 u+1=5(u+6)-u$$

Short Answer

Expert verified
u = 2

Step by step solution

01

Distribute and simplify

Distribute the constants inside the parentheses:For the left side: For the right side:
02

Combine like terms

Combine the terms involving the variable and the constant terms on both sides of the equation:
03

Move all variable terms to one side

Get all the terms involving by subtracting it on both sides:
04

Solve for

After simplifying, isolate by subtracting the constant term from both sides:

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations is about finding the value of the variable that makes the equation true. It involves a series of strategic algebraic steps. For every equation, we want to make both sides equal by performing operations that simplify and balance the equation. These operations include addition, subtraction, multiplication, and division. Each step should maintain the balance of both sides of the equation.
Combining Like Terms
Combining like terms means adding or subtracting terms that have the same variable component. For example, in the equation given: \(6(u-1) + 5u + 1 = 5(u+6) - u\), \(6u\) and \(5u\) are like terms on the left side. We can combine them to make the equation simpler.

When combining like terms, focus on:
  • Grouping terms that have the same variable.
  • Adding or subtracting their coefficients.
For example, \(6u + 5u\) becomes \(11u\). It's a way to streamline the problem, which makes solving it easier.
Distributive Property
The distributive property is crucial for simplifying expressions inside parentheses. It involves multiplying a single term by each term inside the parentheses. For the equation \(6(u-1)\):
  • Multiply \(6\) by \(u\) to get \(6u\).
  • Multiply \(6\) by \(-1\) to get \(-6\).
The same applies to the right side with \(5(u+6)\):
  • Multiply \(5\) by \(u\) to get \(5u\).
  • Multiply \(5\) by \(6\) to get \(30\).
Using the distributive property makes the equation easier to work with by eliminating parentheses.
Isolating Variables
Isolating the variable means getting the variable by itself on one side of the equation.In the equation \(11u - 5u + 1 - 7 = 0\), the variables are combined and simplified first:
  • \(6u - u = 11u\).
  • On the right side, combining terms \(-u\) gives \(5u - u = 4u\).
Next, move all variable terms to one side of the equation. For instance:
  • Subtract \(4u\) from both sides to get \(7u = 35\).
Finally, divide both sides by \(7\) to solve for \(u\). Isolating the variable by performing these steps will help in arriving accurately at the solution.

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