Problem 63 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. \(7+2(3 x-5)=8-3(2 x+1)\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{3}{4}\)

Step 1: Apply Distributive Property

Distribute the constants before the parentheses on both sides of the equation: \(7+6x-10=8-6x-3\)

Step 2: Simplify Both Sides

Simplify both sides of the equation by combining like terms. This will yield: \(-4x=-3\)

Step 3: Solve for x

Now divide both sides of the equation by -4 to solve for 'x'. This will yield: \(x = \frac{3}{4}\)

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that allows us to simplify expressions and solve equations more effectively. This property involves distributing a multiplied factor across terms inside parentheses. To put it simply: if you see an expression like

• \( a(b + c) \),
• you can "distribute" \( a \) to get \( ab + ac \).

This is quite helpful in making complex expressions more manageable, especially when solving equations.
In our original problem, we start with the equation: \(7 + 2(3x - 5) = 8 - 3(2x + 1) \). The distributive property lets us break down this expression by multiplying the constants (\(2\) and \(-3\)) into the parentheses:

• On the left side, \(2(3x - 5)\) becomes \(6x - 10\).
• On the right side, \(-3(2x + 1)\) becomes \(-6x - 3\).

This step is crucial for simplifying the equation to solve for \(x\). Knowing how to use the distributive property can speed up the problem-solving process and help avoid errors.

Like Terms

Combining like terms is another essential skill in algebra that makes solving equations much easier. Like terms are terms in an expression that contain the same variable raised to the same power.
In simpler terms, you can only combine those terms that "look alike." For example:

• \(3x\) and \(5x\) are like terms because they both contain the variable \(x\).
• In contrast, \(3x\) and \(2y\) are not like terms, due to different variables.

In the original equation, after applying the distributive property, we have: \(7 + 6x - 10 = 8 - 6x - 3\).
We simplify each side by combining like terms.

• On the left: \(7 - 10 + 6x\) becomes \(6x - 3\).
• On the right: \(8 - 3 - 6x\) becomes \(5 - 6x\).

This simplification is essential in reducing the equation to a more solvable form, where you can then isolate the variable \(x\). Mastery of identifying and combining like terms is a key step in making algebra problems more straightforward to solve.

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes integers, fractions, and decimals—basically all the numbers we typically use in everyday life, except for imaginary numbers.
In the context of solving algebraic equations, real numbers are crucial as they represent the possible solutions:

• If an equation is true for all real numbers, it means that any number can satisfy it.
• If it has no solution, there is no real number that makes the equation true.

In our example, after simplifying the equation to \(6x - 3 = 5 - 6x\), we solve and find that \(x = \frac{3}{4}\) is the solution. This tells us there is a specific real number for which the original equation holds true.
Understanding the role of real numbers helps us identify whether our equations are valid and whether they can be solved within the realm of the numbers we're familiar with. This is why solutions in algebra are often described in terms of real numbers, ensuring clarity about the scope of possible answers.