Problem 49 - Elementary Algebra Within Reach

Question

In Exercises 49 and 50 , justify each step of the equation. Then identify any properties of equality used to solve the equation. $$ \begin{aligned} \frac{x}{3} &=x+1 \\ 3\left(\frac{x}{3}\right) &=3(x+1) \\ x &=3 x+3 \\ x-3 x &=3 x+3-3 x \\ x-3 x &=3 x-3 x+3 \\ -2 x &=3 \\ \frac{-2 x}{-2} &=\frac{3}{-2} \\ x &=-\frac{3}{2} \end{aligned} $$

Step-by-Step Solution

Answer

Properties of equality used in this solution include the multiplication and division properties. The solution is \( x = -\frac{3}{2} \).

Step 1: Analyze the first equality

The first line of the equation is simple division, where \( \frac{x}{3} = x + 1 \). This is the equation that needs to be solved.

Step 2: Apply the multiplication property of equality

The second line multiplies both sides of the equation by 3. This gives us a new equation: \(x = 3x + 3\). This operation is justified by the multiplication property of equality which states: if a = b, then a * c = b * c.

Step 3: Simplify by subtracting 3x from both sides

The sucessive lines simplify the equation by subtracting \(3x\) from each side to create a new equation: \(x - 3x = 3x - 3x + 3\) which simplifies to \(-2x = 3\). This operation is justified by the distributive property, which allows removal/subtraction from each side.

Step 4: Divide by -2

The last lines divide both sides of the equation by -2, which results in \(x = -\frac{3}{2}\). This step is justified by the division property of equality: if a = b and c ≠ 0, then \( \frac{a}{c} = \frac{b}{c} \). This step gives us the solution for x.

Key Concepts

Properties of Equality

When solving equations, the properties of equality are fundamental rules. These properties ensure that you can perform the same operation on both sides of the equation without changing its structure or meaning.
Some core properties used in solving equations include:

• Reflexive Property: States that a value is always equal to itself, for example, \( a = a \).
• Symmetric Property: If \( a = b \), then \( b = a \).
• Transitive Property: If \( a = b \) and \( b = c \), then \( a = c \).
• Addition/Subtraction Properties: If you add or subtract the same number from both sides of the equation, the equality remains true. For example, if \( a = b \), then \( a + c = b + c \) or \( a - c = b - c \).

Understanding these helps maintain the equation's balance, ensuring any operations performed do not alter the fundamental truth of the statement.

Multiplication Property of Equality

The multiplication property of equality states that you can multiply both sides of an equation by the same non-zero value, and the equality is preserved. This is highly useful when dealing with fractions or coefficients that you wish to remove from one side of the equation.
For example, in the equation \( \frac{x}{3} = x + 1 \), multiplying both sides by 3—as shown in the second step of our solution—helps eliminate the fraction. This action produces the equation \( x = 3x + 3 \), which is simpler to work with. Just remember, this property is only valid when multiplying by non-zero numbers, since multiplying by zero would make both sides equal to zero and potentially lose the relationship in your original equation.
Therefore, this property allows us to clear out fractions or simplify expressions for easier manipulation, maintaining the balance between both sides of the equation.

Distributive Property

The distributive property is an essential algebra principle that connects addition and multiplication. It explains that multiplying a number by a sum or difference is the same as doing each multiplication separately and then adding or subtracting the results. Mathematically, it's expressed as \( a(b + c) = ab + ac \).
In the context of equation solving, the distributive property can often be used to simplify expressions. For instance, if you perform subtraction or factor out common terms on both sides of the equation, this property allows streamlined simplification.

• In our solution, subtracting \( 3x \) from both sides—\( x - 3x = 3x - 3x + 3 \) which simplifies to \( -2x = 3 \)—demonstrates the use of balancing operations evenly by this property.

This reveals clearer terms and isolates desired variables, which is helpful in solving algebraic equations efficiently.

Division Property of Equality

The division property of equality is significant for isolating variables in an equation. It states that if you divide both sides of an equation by a non-zero number, the equality remains true. This is key for solving equations for a particular variable.
Consider the equation \( -2x = 3 \). To isolate \( x \), we divide both sides by \(-2\), resulting in \( x = -\frac{3}{2} \). This division ensures the variable is expressed as a single entity, showing the exact solution where the variable equals a specific value.
Always remember that the divisor cannot be zero, as division by zero is undefined. Using this property lets you break down complex equations and make the terms manageable, ultimately leading to finding the solution more straightforwardly.