Problem 127 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change \((s)\) to produce a true statement. If \(a\) and \(b\) are any real numbers, then \(a x+b=0\) always has one number in its solution set.

Step-by-Step Solution

Answer

The given statement is false. The true statement would be 'If \(a\) and \(b\) are any real numbers, and \(a\) is not equal to zero, then \(a x+b=0\) always has one number in its solution set.'

Step 1: Identify the Linear Equation

From the statement, we have a linear equation in one variable, \(a x+b=0\), where \(a\) and \(b\) are any real numbers.

Step 2: Check for the Existence of a Solution

For a linear equation in one variable, a common solution exists when the coefficient of the variable is not zero. In this case, if \(a\) (the coefficient of \(x\)) is not zero, the equation will have one unique real number as a solution.

Step 3: Validate Given Statement

Assuming \(a\) and \(b\) to be any real numbers, if \(a=0\) and \(b\neq0\), the equation becomes \(0x+b=0\) or \(b=0\). This indicates no solution because all real values of \(x\) do not satisfy the equation, contradicting the given statement.

Step 4: Correct the Statement

The false statement 'If \(a\) and \(b\) are any real numbers, then \(a x+b=0\) always has one number in its solution set.' can be made true by stating 'If \(a\) and \(b\) are any real numbers, and \(a\) is not equal to zero, then \(a x+b=0\) always has one number in its solution set.'

Key Concepts

Understanding Real Numbers

Real numbers are fundamental in mathematics. They include both rational and irrational numbers. Rational numbers are those that can be expressed as a fraction, while irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimals.
Together, real numbers cover all numbers that can be found on the number line, from negative infinity to positive infinity.

• Rational numbers include integers, fractions, and repeating or terminating decimals.
• Irrational numbers include numbers like \(\pi\) and \(\sqrt{2}\), which cannot be precisely written as fractions.

In the context of linear equations like \(a x + b = 0\), the coefficients \(a\) and \(b\) can be any real number. This means they can be integers, fractions, square roots, and many other forms.
Understanding real numbers helps in manipulating equations correctly and predicting solutions accurately.

Exploring the Solution Set

The solution set of a linear equation refers to the set of all possible values of the variable that satisfy the equation. For a simple linear equation like \(a x + b = 0\), this typically means finding the value of \(x\) that makes the equation true.

• If the equation is \(2x + 3 = 0\), the solution \(x = -\frac{3}{2}\) is part of the solution set as it satisfies the equation.
• If an equation cannot be satisfied by any real number, it is said to have no solution, meaning its solution set is empty.

Consider the mentioned scenario: If \(a=0\) and \(beq0\), the equation becomes \(0x + b = 0\) or simply \(b = 0\). Here, the solution set is empty because no real value of \(x\) will satisfy \(beq0\).
On the other hand, when \(aeq0\), the solution set contains exactly one unique value for \(x\).

Unique Solutions in Linear Equations

In linear equations, a unique solution means there is exactly one value of the variable that satisfies the equation. For \(a x + b = 0\), if \(aeq0\), there will be one and only one value of \(x\) that can solve the equation. This makes the solution unique.

• The formula to find this unique solution is \(x = -\frac{b}{a}\), assuming \(aeq0\).
• This unique value exists because dividing \(-b\) by \(a\) gives a single value on the number line.

Unique solutions are crucial in determining the behavior of equations and predicting outcomes in real-world applications. Whenever \(a\) is not zero, it ensures that the line represented by the equation intersects the x-axis at a single, distinct point.
However, if \(a = 0\), the situation changes drastically, and the equation might not even have a solution. This highlights why having \(aeq0\) is essential for ensuring a unique solution exists.