Problem 35 - Algebra Form and Function

Question

Does the equation have no solution, one solution, or an infinite number of solutions? $$ 4 x+3=4(x-1)+7 $$

Step-by-Step Solution

Answer

Answer: The equation holds true for all values of x, and therefore, there is an infinite number of solutions.

Step 1: Distribute the constants on both sides of the equation

Multiply 4 by the expression (x - 1) in the equation: $$ 4x+3=4(x-1)+7 $$ Which gives: $$ 4x+3=4x-4+7 $$

Step 2: Combine like terms

Combine the constants on the right side of the equation: $$ 4x+3=4x+3 $$

Step 3: Check for solutions

Notice that both sides of the equation are identical, meaning that there are no differences between the two sides. This indicates that the equation holds true for all values of x, and therefore, there is an infinite number of solutions.

Key Concepts

Infinite Solutions

When an equation has infinite solutions, it means that any value for the variable will satisfy the equation. In other words, every possible number you plug into the equation will make it true. This occurs when, after simplifying an equation, both sides are identical.
In the exercise, after simplifying, we ended up with the equation \(4x + 3 = 4x + 3\). Since both sides are exactly the same, it means that no matter what value \(x\) takes, the equation will always be true.

• An equation with infinite solutions is also known as an identity.
• It showcases complete equivalence between both sides of the equation.

Recognizing infinite solutions is essential as it tells us there isn't just one value of \(x\) that works, but instead all values do.

Distributive Property

The distributive property is a fundamental principle in algebra that helps simplify expressions by distributing multiplication over addition or subtraction inside parentheses. When you apply this property, you multiply a term outside the parentheses by each term inside the parentheses.
In the given exercise, the distributive property is used to multiply 4 by the expression \((x - 1)\), resulting in \(4x - 4\). This step helps in breaking down complex expressions into simpler parts:

• The expression \(a(b + c)\) becomes \(ab + ac\).
• This property is crucial for solving equations as it allows for the simplification of terms.

Using the distributive property correctly ensures that all terms are accurately accounted for as you solve for the variable.

Combining Like Terms

Combining like terms is an essential algebraic process that involves merging terms with the same variable raised to the same power. It helps in simplifying polynomial expressions and solving equations efficiently.
In our example, after applying the distributive property, we combined like terms on the right side of the equation. The terms \(-4\) and \(+7\) were combined to yield \(+3\), resulting in the equation \(4x + 3 = 4x + 3\).

• Terms are considered 'like' if they have the same variable and exponent.
• By combining them, equations can be simplified, making them easier to solve.

Understanding how to combine like terms aids in reducing complexity, guiding you to the solution more clearly and quickly.