Problem 43 - Elementary Algebra

Question

For the following problems, solve the linear equations in two variables. $$ y=0(3 x+9)-1, \text { if } x=12 $$

Step-by-Step Solution

Answer

Answer: -1

Step 1: Substitute the given value of x into the equation

We are given that x = 12. We need to substitute this value into the equation: $$ y = 0(3\cdot 12 +9)-1 $$

Step 2: Perform the arithmetic operations

Now we need to perform the arithmetic operations in the equation in the proper order - first calculate the value in the parentheses, then multiply by the scalar, and finally subtract 1: $$ y = 0(3\cdot 12 +9)-1 \\ y = 0(36 + 9)-1 \\ y = 0(45)-1 $$

Step 3: Simplify the equation

Next, we need to multiply 0 by 45 and then subtract 1: $$ y = 0\cdot 45 - 1 \\ y = 0 - 1 $$

Step 4: Determine the value of y

Finally, we need to subtract 1 from 0: $$ y = -1 $$ Thus, when x = 12, the value of y in the given linear equation is -1.

Key Concepts

Two Variables

Linear equations often involve two different types of variables. In the exercise given, these variables are \(x\) and \(y\). Each variable represents a different quantity.
Understanding how these variables interact with each other is key to solving the equations.

• The variable \(x\) often represents what is known as an independent variable. It's a variable you can change or control.
• The variable \(y\), on the other hand, is typically the dependent variable. Its value depends on \(x\).

In the problem stated, you are given a value for \(x\) (i.e., 12) and need to find \(y\). Linear equations with two variables like this often form a straight line when graphed on a coordinate plane, demonstrating the relationship between \(x\) and \(y\).
The relationship is such that modifying \(x\) affects \(y\). Understanding this interaction is crucial for mastering linear equations.

Substitution Method

The substitution method is a technique used to find the values of variables in equations with two variables. It's especially helpful when you have one variable expressed in terms of the other. In this method, you solve one of the equations for one variable and substitute this expression into another equation.
The exercise provided an already simplified equation for substitution.

• Here, you substitute the given \(x = 12\) into the equation \(y = 0(3x + 9)-1\).
• This involves replacing \(x\) with 12 to simplify the expression for \(y\).

Substitution helps in reducing complex multi-variable problems into simpler single-variable equations, making it easier to solve.

Arithmetic Operations

Arithmetic operations are fundamental processes in solving equations. In the problem, you need to handle several operations like multiplication, addition, and subtraction in a specific order.
Following the proper order is crucial:

• Start with operations inside the parentheses, known as the order of operations or PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• First, solve \(3\times 12 + 9 \).
• Next, multiply by 0, which often eliminates any term (since zero times any number is zero).
• Finally, address the subtraction: \(0 - 1\).

Understanding and implementing arithmetic order correctly ensures you simplify equations accurately.

Solving Linear Equations

Solving linear equations involves finding the unknown variables. Once you substitute and perform necessary operations, simplifying the equation will yield the solution. In this exercise:

• The equation simplifies from \(y = 0 \cdot (45) - 1\) to \(y = 0 - 1\).
• This results in \(y = -1\).

Effectively solving linear equations often builds a strong foundation for handling more complex algebraic expressions. The ultimate goal is to isolate the variable you are solving for. You accomplish this through careful arithmetic manipulation, guided by the equation's inherent structure and operations.