Problem 8 - Algebra 1

Question

Jan says her work shows that 6 is not a solution of \(3 x-4=14 .\) What is a likely explanation for her error?

Step-by-Step Solution

Answer

Jan's error may have been a miscalculation or misunderstanding of the problem. The correct calculation shows that 6 is indeed a solution to the equation \(3x - 4 = 14\).

Step 1: Substitute the value of x

According to Jan, 6 is not a solution of the equation \(3x - 4 = 14\). Let's substitute \(x = 6\) into the equation.

Step 2: Evaluate the left-hand side of the equation

On substituting the value of x into the equation we get \(3*6 - 4\) which equals to 14.

Step 3: Compare the evaluated expression with the right-hand side of the equation

The left hand side of the equation is 14, which is equal to the right hand side of the equation.

Step 4: Draw a conclusion

Since the evaluated expression is equal to the right side of the equation, we can conclude that 6 is indeed a solution of the equation. Therefore, Jan's error is likely due to a misunderstanding or a computational mistake in her work.

Key Concepts

Substitution Method

The substitution method is a technique used to determine if a specific number is a solution to an equation, especially when dealing with linear equations like \(3x - 4 = 14\). To apply this method, you simply replace the variable with the proposed solution number. For instance, if you are checking whether \(x = 6\) solves the equation, you substitute 6 in place of \(x\). You then calculate the left-hand side of the equation using this number. This method is widely used because it's straightforward and helps in verifying solutions quickly. Just remember: substitute wisely and solve it like a puzzle— piece by piece, until the pieces fit.

Algebraic Expressions

Algebraic expressions are mathematical statements that include numbers, variables, and operators like addition or multiplication. They form the building blocks of equations. For example, \(3x - 4\) is an algebraic expression.In an algebraic expression, variables like \(x\) can represent unknown values. The expression evaluates to different numbers depending on the variable's value you choose. To solve an equation, you often need to simplify or manipulate algebraic expressions to isolate the variable, finding what value makes the entire equation true. Understanding how these work is essential as they are pivotal to solving more complex problems.

Equation Balancing

Equation balancing is crucial when solving linear equations. When you manipulate an equation, it's important to keep both sides equal to maintain the equation's validity. Whenever you perform an operation, like addition or multiplication, on one side of an equation, you must perform the same operation on the other side. This ensures that the equation stays balanced. Think of balancing an equation like balancing a seesaw: whatever you do on one side must be mirrored on the other side to keep it from tipping. In the equation solving process, balancing helps to isolate terms and ensures you derive the correct solution without errors.