Problem 48 - Intermediate Algebra

Question

Replace each question mark to make the given statement an application of the indicated property of equality. For example, \(16=\) ? becomes \(16=16\) because of the reflexive property of equality. If \(t=4\) and \(s+t=9\), then \(s+?=9\) (Substitution property of equality)

Step-by-Step Solution

Answer

Replace the question mark with 4 in the equation, so it becomes \(s+4=9\).

Step 1: Identify Known Values

We are given two equations: \(t=4\) and \(s+t=9\). Identify what value can be substituted because of the given information.

Step 2: Apply Substitution Property

The substitution property of equality states that if two things are equal, one can replace the other in an expression or equation. Since \(t=4\), substitute \(4\) for \(t\) in the equation \(s+t=9\). This gives us \(s+4=9\).

Step 3: Verify Substituted Equation

Check that the substitution leads to a valid equation: substitute \(4\) into the equation for \(t\) to verify that \(s+4=9\) is consistent with the condition that \(s+t=9\).

Key Concepts

Substitution Property

The substitution property of equality is a fundamental tool in algebra that allows one to replace a variable or a term in an equation or expression with another.
If you know two things are equal, you can substitute one for the other. For instance, if you have \(t = 4\), and you also know \(s + t = 9\), you can use the substitution property.
In this context, because \(t\) is equal to 4, you can substitute 4 in place of \(t\) in the second equation.

• Take the equation \(s + t = 9\).
• Since \(t = 4\), replace \(t\) with 4: \(s + 4 = 9\).

By substituting, you simplify the expression or equation and make it easier to solve for the unknown variables. This approach is consistent and ensures that your equations remain balanced, honoring the initial equalities given in the problem.

Reflexive Property

The reflexive property of equality is quite straightforward. It tells us that any number or expression is always equal to itself. It's like saying "a" is always equal to "a". This property ensures that the equality is always balanced and initial assumptions about equality hold.

• For example, the statement \(16 = 16\) is an application of the reflexive property because the number on both sides of the equal sign is the same.
• Similarly, if you have a variable expression like \(x = x\), it still illustrates the reflexive property because \(x\) is equal to itself.

This property is crucial in mathematics because it sets the foundation for affirmations of equal values before any further operations are conducted. Knowing this is one of the base principles helps establish the validity of other equations you'll solve.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division.
They can represent single arithmetic problems or part of larger equations.

• A simple algebraic expression could be \(3x + 4\), where \(3x\) means 3 times the value of x.
• Expressions like \(s + 4\) in our exercise are important building blocks in forming equations.

In understanding expressions, it's crucial to recognize that they do not have equality signs (unlike equations) and thus do not state relationships by themselves. To solve for variables using expressions, you typically convert them into equations, establishing relationships between known and unknown quantities. This approach helps in simplifying and solving various algebraic problems, especially when combined with other properties like substitution and reflexivity.