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Chapter 2: Problem 65

Solve each equation, if possible. $$ 5 x=4 x+7 $$

Short Answer

Expert verified
The solution is \(x = 7\).

Step by step solution

01

Identify the Equation Type

The given equation is a linear equation. This means that it is an equation of the first degree, involving only the variable \(x\) with no exponents other than 1.
02

Rearrange the Equation

To solve for \(x\), aim to isolate \(x\) on one side of the equation. Start by subtracting \(4x\) from both sides to eliminate \(x\) terms on the right side:\[ 5x - 4x = 4x - 4x + 7 \]This simplifies to:\[ x = 7 \]
03

Verify the Solution

Substitute \(x = 7\) back into the original equation to verify the solution. Start with the left-hand side (LHS) and right-hand side (RHS) of the original equation:- LHS: \( 5x = 5(7) = 35 \)- RHS: \( 4x + 7 = 4(7) + 7 = 28 + 7 = 35 \)Since LHS equals RHS, the solution \(x = 7\) is verified.

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Key Concepts

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

Solving Equations
When we talk about solving equations, especially linear ones, we are trying to find the value of the variable that makes the equation true. Linear equations are the simplest type of equation and they appear in the form of \(ax + b = 0\), where \(a\) and \(b\) are constants. In our example, the equation is \(5x = 4x + 7\). Here, both sides of the equation contain the variable \(x\), which makes the first step moving all terms containing \(x\) to one side. By doing this, we're streamlining the equation, preparing it for the next steps.
Variable Isolation
The concept of variable isolation revolves around getting the variable, \(x\), alone on one side of the equation. This is crucial in providing a clear pathway to the solution. Let's take a look at our example equation again: \(5x = 4x + 7\). To isolate \(x\), subtract \(4x\) from both sides:
  • Start with the original equation: \(5x = 4x + 7\).
  • Subtract \(4x\) from both sides: \(5x - 4x = 4x - 4x + 7\).
  • This simplifies to \(x = 7\).
By isolating \(x\), we've directly found that \(x = 7\). This step is key in solving any linear equation, as it paves the way for clarity and simplicity in identifying the value of the variable.
Equation Verification
Verification in solving equations is all about ensuring that our solution is correct. This involves plugging the solution back into the original equation to see if both sides (left-hand side LHS and right-hand side RHS) equal each other. Let's verify our solution for \(x = 7\) from the example:
  • Start with the LHS: \(5x = 5(7) = 35\).
  • Then check the RHS: \(4x + 7 = 4(7) + 7 = 28 + 7 = 35\).
  • If both LHS and RHS match, as they do here (both are 35), then our solution \(x = 7\) is correct.
Verification gives us confidence in our solution and confirms the integrity of the steps we followed in solving the equation.

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