Problem 75 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$5 t-4+3 t=8 t-4$$

Step-by-Step Solution

Answer

The solution to the equation is \( t=\) all real numbers.

Step 1: Simplify the Equation

Combine like terms on both sides of the equation. This gives us \(8t-4=8t-4\)

Step 2: Subtract 8t from Both Sides

Subtract \(8t\) from both sides to leave the variable on one side of the equation and the constant on the other which results in \( -4=-4\)

Step 3: Inteprate the result

Since both sides of the equation are equal, it means the given equation is true for all real numbers.

Step 4: Check your Solution

One can easily see that for all real numbers, the equation is satisfied. Thus, the solution to the equation is \( t=\) all real numbers.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols, such as addition, subtraction, multiplication, and division. These expressions do not have an equal sign, unlike an equation which must equate two expressions. In the exercise provided, the parts of the equation before it's solved like \(5t - 4 + 3t\) are algebraic expressions. They consist of numbers and a variable, \(t\), combined using arithmetic operations.

Algebraic expressions can be simplified by combining like terms, which are terms that have the same variable raised to the same power. In solving the equation, combining \(5t\) and \(3t\) to form \(8t\) is a typical simplification of algebraic expressions. Such simplifications are crucial when solving algebraic equations as they help make the equation easier to solve by reducing the number of terms.

Real Numbers

Real numbers include all the numbers on the number line. This encompasses rational numbers like integers (e.g., -1, 0, 1), fractions (e.g., 1/2, -3/4), and irrational numbers like \(\pi\) and \(\sqrt{2}\). In the context of the exercise, the solution indicates that \(t\) could be any real number because the equation holds true regardless of what real number \(t\) takes.

Understanding real numbers is fundamental in solving such equations because it indicates the potential solutions are not limited to integers or fractions alone. The conclusion that "the equation is true for all real numbers" typically follows when simplifying an expression shows identical truths on both sides of an equation, as in our exercise with \(-4 = -4\). This means any real number you substitute for \(t\) still makes the equation valid.

Like Terms

Like terms are terms in an algebraic expression that have the same variables raised to the same power. Only like terms can be combined or simplified within an expression. For example, in the equation \(5t - 4 + 3t = 8t - 4\), the terms \(5t\) and \(3t\) are considered like terms because they both contain the variable \(t\) to the first power.

When solving equations, combining like terms is one of the first steps because it simplifies the expressions and reduces the complexity of the equation. This can make the solving process more manageable. In our exercise, merging \(5t\) and \(3t\) to form \(8t\) is a demonstration of this fundamental concept. Recognizing and combining like terms is essential for breaking down equations into simpler parts, which is a vital skill in algebra.