Problem 18 - Algebra for College Students

Question

Which one of the following linear equations does not have solution set \(\\{\) all real numbers \(\\} ?\) A. \(4 x=5 x-x\) B. \(3(x+4)=3 x+12\) C. \(4 x=3 x\) D. \(\frac{3}{4} x=0.75 x\) E. \(4(x-2)=2(2 x-4)\) F. \(2 x+18 x=20 x\)

Step-by-Step Solution

Answer

Equation C has a single solution, while all others hold for all real values of x.

Step 1 - Analyze Equation A

Simplify A: \(4x = 5x - x\). Combine like terms on the right side: \(4x = 4x\). This equation holds for all real values of x.

Step 2 - Analyze Equation B

Simplify B: \(3(x+4) = 3x + 12\). Distribute 3: \(3x + 12 = 3x + 12\). This equation holds for all real values of x.

Step 3 - Analyze Equation C

Simplify C: \(4x = 3x\). Subtract \(3x\) from both sides: \(x = 0\). This equation has only one solution.

Step 4 - Analyze Equation D

Simplify D: \(\frac{3}{4}x = 0.75x\). Convert \(0.75\) to a fraction: \(\frac{3}{4}x = \frac{3}{4}x\). This equation holds for all real values of x.

Step 5 - Analyze Equation E

Simplify E: \(4(x-2) = 2(2x-4)\). Distribute: \(4x - 8 = 4x - 8\). This equation holds for all real values of x.

Step 6 - Analyze Equation F

Simplify F: \(2x + 18x = 20x\). Combine like terms: \(20x = 20x\). This equation holds for all real values of x.

Key Concepts

Linear Equations

Linear equations are mathematical statements where each term is either a constant or a product of a constant and a single variable. These equations form straight lines when graphed on a coordinate plane. The standard form of a linear equation in one variable is usually written as: \[ax + b = 0\]where \(a\) and \(b\) are constants, and \(x\) is the variable. Linear equations can have one solution, infinitely many solutions, or no solution at all. For instance, the linear equation \(4x = 4x\) holds true for all real numbers. Conversely, the equation \(4x = 3x\) simplifies to \(x = 0\), which means it has exactly one solution.

Solution Set

The solution set of a linear equation comprises all the values of the variable that make the equation true. When we say a linear equation has a solution set of 'all real numbers,' it means that any real number substituted into the variable will satisfy the equation. For example, the equation \(3(x + 4) = 3x + 12\) simplifies to \(3x + 12 = 3x + 12\), which holds true for any value of \(x\). Conversely, the solution set for \(4x = 3x\) is just \{0\} because only \(x = 0\) makes the equation true.

Real Numbers

Real numbers encompass all the numbers on the number line, including both rational and irrational numbers. These numbers can be positive, negative, or zero. When dealing with linear equations, we often solve for real numbers. For example, a solution that includes 'all real numbers' means that any number you can think of, including fractions and square roots, will satisfy the equation. In the earlier example of \(4x = 4x\), any real number substituted for \(x\) makes the equation true.

Simplification

Simplification is a process in algebra where we make an equation easier to solve by performing operations such as combining like terms and using the distributive property. Simplifying an equation helps in identifying whether it has one solution, no solution, or infinitely many solutions. For instance, when simplifying \(4(x - 2) = 2(2x - 4)\), we distribute the constants: 4x - 8 = 4x - 8Since the left and right sides are identical, this equation is true for all values of \(x\). On the other hand, for \(4x = 3x\), subtracting \(3x\) from both sides simplifies it to \(x = 0\), which is a single solution.