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Chapter 2: Problem 11

Solve the equations by using the subtraction property. $$x+4=7$$

Short Answer

Expert verified
The solution to the equation \( x + 4 = 7 \) is \( x = 3 \).

Step by step solution

01

Identify the Equation

The equation we have is \( x + 4 = 7 \). Our goal is to solve for \( x \), which means we need to isolate \( x \) on one side of the equation.
02

Apply the Subtraction Property

To isolate \( x \), we need to eliminate the \(+4\) from the left side of the equation. We do this by subtracting \( 4 \) from both sides of the equation: \[ x + 4 - 4 = 7 - 4 \] Applying this subtraction gives us a new equation: \( x = 3 \).
03

Verify the Solution

To ensure our solution is correct, substitute \( x = 3 \) back into the original equation: \[ (3) + 4 = 7 \] This simplifies to \( 7 = 7 \), which is correct. Therefore, our solution is verified.

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Key Concepts

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

Subtraction Property
The subtraction property of equality is a fundamental concept in solving linear equations. It states that if you subtract the same number from both sides of an equation, the equality will still hold. This is crucial because it allows us to manipulate equations to better isolate the variable. For example, in the equation \( x + 4 = 7 \), our goal is to isolate \( x \). Here is where the subtraction property becomes helpful:
  • We subtract \( 4 \) from both sides of the equation.
  • This action can be expressed as \( x + 4 - 4 = 7 - 4 \).
  • After performing the subtraction, we have \( x = 3 \).
By using the subtraction property, we successfully isolated the variable, bringing us a step closer to finding the value of \( x \). Always remember, the same operation must be performed on both sides to maintain equality.
Isolation of Variable
The isolation of a variable is a key step in solving equations. It refers to the process of getting the variable alone on one side of the equation. This makes it easier to identify its value. Let's see how it works using our example equation:
  • Starting with \( x + 4 = 7 \), you need to get \( x \) by itself.
  • We notice \( 4 \) is added to \( x \), so we need to remove this \( 4 \) from the left side.
  • Using the subtraction property, we achieve isolation by subtracting \( 4 \) from both sides, resulting in \( x = 3 \).
Now, the variable \( x \) is isolated. Once isolated and simplified, you can easily notice that \( x \) equals 3. This makes it evident that the solution meets the requirements of the equation. The isolation of the variable is essential for a clear and straightforward solution.
Verification of Solution
Verifying the solution of an equation is an important final step to ensure the solution is correct. By substituting the value back into the original equation, you can check your work:
  • Take the equation \( x + 4 = 7 \), and replace \( x \) with \( 3 \), the value we calculated.
  • The substitution becomes \( 3 + 4 \).
  • After simplifying, we find that \( 3 + 4 = 7 \).
The original equation is satisfied because both sides equal 7. This indicates that our solution \( x = 3 \) is indeed correct. Verification is a vital step, especially when learning to solve equations, as it confirms the correctness of your answer and builds confidence in your problem-solving abilities.

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Most popular questions from this chapter

Consider the formula $$\frac{x-50}{\frac{s}{\sqrt{n}}}$$ Evaluate this expression using the given values. Round the results to two decimal places. $$x=50.5, s=6.3, n=30$$

Consider the formula $$\frac{0.55-p}{\sqrt{\frac{p(1-p)}{n}}}$$ Evaluate this expression using the given values. Round the results to three decimal places. $$p=0.6, n=41$$

Simplify the expressions, which involve exponents and square roots. Round the results to two decimal places as necessary. $$\frac{(1.65)^{2} \cdot 0.25}{0.05^{2}}$$

Consider the formula $$\frac{x-50}{\frac{s}{\sqrt{n}}}$$ Evaluate this expression using the given values. Round the results to two decimal places. $$x=45.6, s=4.4, n=45$$

Apply the order of operations and answer the questions. The volume of a certain bucket, in cubic inches, is approximated by the expression \(\pi \cdot 4.43^{2} \cdot(15.8) .\) Evaluate the expression and interpret the result. Round the volume to two decimal places.
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