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Chapter 8: Problem 126

Solve the equation for the variable. $$4 x+36=10 x$$

Short Answer

Expert verified
x = 6

Step by step solution

01

Move all terms involving the variable to one side

Subtract \(4x\) from both sides of the equation to isolate the variable terms on one side. This gives us:\[4x + 36 - 4x = 10x - 4x\]Simplifying this, we get:\[36 = 6x\]
02

Solve for the variable

Divide both sides by 6 to solve for \(x\):\[\frac{36}{6} = \frac{6x}{6}\]Simplifying this, we get:\[x = 6\]

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Key Concepts

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

isolation of variables
Isolation of variables means getting the variable you're solving for on one side of the equation by itself. In the equation given, we started with \[4x + 36 = 10x\]. To isolate the variable term, you need to first move all the terms involving the variable to one side.
This is done by subtracting or adding terms from both sides of the equation.
In our example, we subtracted \[4x\] from both sides, which helped us isolate the variable terms on one side. Let's see this step again:\[4x + 36 - 4x = 10x - 4x\]. This simplifies to:\[36 = 6x\]. Notice how all the \[x\] terms are now on the right side while the constant is on the left side, achieving isolation of the variable.
simplifying equations
Simplifying equations involves combining like terms and reducing the equation to its simplest form. This process is essential to make solving the equation easier. In our given equation, after isolating the variable, we had:\[36 = 6x\]. This equation is already simplified because there are no like terms to combine anymore. However, in many problems, you may need to combine like terms, distribute multiplication over addition, or factor terms to simplify the equation. Here are a few tips for simplifying equations:
  • Combine like terms on both sides of the equation.
  • Use the distributive property if necessary, which means \[a(b + c) = ab + ac\].
  • Clear fractions by multiplying through by a common denominator.
Simplifying makes the equation more straightforward and easier to solve.
basic algebra steps
Solving linear equations often requires following some basic algebra steps. These are essential to reach the solution correctly. Using our equation\[4x + 36 = 10x\], let's break down these steps further:
  • Step 1: Isolate the Variable. Start by getting all terms with the variable on one side. This often involves adding or subtracting terms on both sides of the equation.
  • Step 2: Combine Like Terms. Simplify the equation by combining all like terms. This makes it easier to work with.
  • Step 3: Solve for the Variable. This step involves performing any arithmetic needed to get the variable by itself. In our example, we divided by 6:\[\frac{36}{6} = \frac{6x}{6}\] Simplifying this, we get\[x = 6\].
Following these steps ensures you methodically solve linear equations without missing any critical part of the process. Knowing these steps by heart will make handling algebra problems much easier.

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