Problem 76 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$7 z-5 z-8=2 z-8$$

Step-by-Step Solution

Answer

The given equation is true for all real numbers, hence, it has infinitely many solutions.

Step 1: Simplify Both Sides

On the left side of the equation, you can combine like terms \( 7z - 5z \) to get \( 2z - 8 \). Meanwhile, the right side of the equation already appears to be in its simplest form. Hence, the simplified equation becomes \( 2z - 8 = 2z - 8 \).

Step 2: Subtract \( 2z \) from Both Sides

To isolate \( z \), you need to eliminate \( 2z \) from one of the sides. By subtracting \( 2z \) from both sides, you get \( -8 = -8 \).

Step 3: Check for Validity

Notice that after eliminating \( z \), you are left with \( -8 = -8 \). This equality is indeed true, meaning that any real number will satisfy the original equation. Since there is no variable to solve, it turns out this was an identity, supporting the existence of infinitely many solutions.

Key Concepts

Like Terms

Combining like terms is a fundamental skill in algebra that makes solving equations easy. Like terms are terms in an equation that have the same variable raised to the same power. For example, in the expression \(7z - 5z\), both terms are like terms because they both involve the variable \(z\) raised to the first power. This means they can be directly added or subtracted.

• Identify Like Terms: Look for terms that have the same variable and exponent.
• Combine Like Terms: Add or subtract their coefficients.

Combining like terms simplifies expressions, making it easier to solve for the variable involved. In our example, combining \(7z\) and \(-5z\) results in \(2z\), which simplifies the left side of the equation.

Identity Equations

An identity equation is a type of equation that holds true for all values of the variable involved. Identifying an identity equation often happens after simplifying both sides and finding that they are identical.In our exercise, after simplifying both sides of the equation, we find that they match exactly: \(2z - 8 = 2z - 8\). This is a clear indication of an identity equation.

• Simplification: Compare both sides of your equation once simplified.
• Identical Sides: If they are the same, then you likely have an identity equation.

Identity equations are unique because they confirm that no matter what value is chosen for \(z\), the equation will always be true.

Infinitely Many Solutions

Equations with infinitely many solutions are those that hold true regardless of the variable value. When you encounter an identity equation, it means that there are infinite solutions. This results because whatever number you substitute for the variable, the equality holds.In our example, after simplifying to find \(-8 = -8\), this tautology shows that the equation is genuine no matter what \(z\) is.

• Recognizing Infinitely Many Solutions: If simplifying results in a true statement without the variable, the equation has infinitely many solutions.
• Understanding: Any real number can be a solution, and every substitution will satisfy the equation.

Always remember, identifying this type of solution involves recognizing that simplified sides are an identity.