/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Solve the given equations. $$6... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 18

Solve the given equations. $$6+4 L=5-3 L$$

Short Answer

Expert verified
The solution is \(L = \frac{-1}{7}\).

Step by step solution

01

Rearrange the equation

Begin by getting all terms with the variable on one side and constant terms on the other. We can do this by adding \(3L\) to both sides of the equation:\[6 + 4L + 3L = 5\] which simplifies to:\[6 + 7L = 5\]
02

Isolate the variable term

Next, we aim to isolate the term with \(L\). Subtract 6 from both sides to move constant term to right:\[7L = 5 - 6\]which simplifies to:\[7L = -1\]
03

Solve for the variable

Now, divide both sides by 7 to solve for \(L\):\[L = \frac{-1}{7}\]

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Key Concepts

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

Variable Isolation
In the journey of solving equations, one of the first major steps is isolating the variable. This is essential to finding what the variable actually represents in the equation. The word 'isolation' means to get the variable by itself on one side of the equation, typically the left.
When you start with an equation like \(6 + 4L = 5 - 3L\), the goal is to manipulate it until "L" is alone on its side of the equation.
Here's how to approach it:
  • Look for terms that can be combined or moved without changing the balance of the equation.
  • Use operations like addition or subtraction to move terms across the equals sign.
  • Rearrange the equation step by step until you see the variable 'L' by itself with its coefficient.
Practicing this method makes isolating variables feel like a natural process rather than a chore.
Algebraic Manipulation
Algebraic manipulation involves using mathematical operations to change the form of an equation. The heart of this lies in applying basic operations correctly to simplify and solve equations.
Manipulation is about understanding that numbers and operations on both sides of the equation are linked by equality. To stay true to this link, you must
  • Apply operations (addition, subtraction, multiplication, division) consistently on both sides.
  • Combine like terms to make the equation simpler. Like in our example, combine \(4L\) and \(3L\) to become \(7L\).
  • Understand and apply the principle of opposites like subtracting \(6\) to peel away terms you want removed or simplified.
This becomes a powerful skill for solving more complicated equations later on.
Equation Balancing
Equation balancing is crucial when solving any algebraic equation. The idea of an equation is that both sides should weigh the same. If you do something to one side, you should do exactly the same to the other.
Consider the equation balancing the act of maintaining equality:
  • Visualize your equation like a balanced scale; any shift you make on one side should be mirrored on the other.
  • Avoid errors by ensuring each step has a mirrored operation, which keeps the equation valid.
  • Double-check your adjustments; mistakes often happen if operations aren鈥檛 evenly applied on both sides.
Through practicing balance, you gain a solid understanding of how to keep equations true and solutions reliable.

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