Problem 4 - Elementary Algebra

Question

\(y=-9 x-14 .\) Determine the value of \(y\) if \(x=-3 .\)

Step-by-Step Solution

Answer

Answer: The value of y is 13 when x = -3 in the linear equation.

Step 1: Write down the given equation and substitute the value of x

We are given the equation \(y = -9x - 14\). We substitute the value of x, which is -3, into the equation: \(y = -9(-3) - 14\).

Step 2: Solve the equation for y

Now, we need to solve for y by performing arithmetic operations: \(y = (-9\times-3) - 14\) \(y = 27 - 14\) \(y = 13\)

Step 3: State the answer

After solving the equation, we found that when \(x = -3\), the value of \(y = 13\).

Key Concepts

Substitution Method

When we come across linear equations, one efficient way to find the unknown variable is through the substitution method. The essence of this method is to replace the value of one variable with its equivalent so that the equation only has one variable left to solve. For instance, imagine you're given the formula for a secret sauce, where the taste factor (T) changes based on the amount of spice (S) you add: T = 2S + 4. If you know the amount of spice you're adding is 3 units, you substitute that into the equation in place of S.

Applying this to our exercise, we're presented with a linear equation where the value of 'x' is already provided. By substituting -3 for 'x' in the equation, we immediately transform it into a simpler problem where we just have to perform basic arithmetic to find 'y'. Remember, the substitution method is a powerful tool in your math toolkit. It's versatile and can simplify complex problems into ones that are much easier to digest and solve.

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics. They include addition, subtraction, multiplication, and division. These operations allow us to combine numbers to obtain new values. In the realm of solving linear equations, arithmetic operations help us simplify the equations to find the solutions.

Let's break down the arithmetic performed in our example. After substituting -3 for 'x', we get \(y = (-9\times-3) - 14\). The multiplication \(\times\) is our first operation, turning \(9\times -3\) into -27. Next, we carry out the subtraction (-), taking -27 from -14, which simplifies to 13. These operations may seem simple, but they play a crucial role in unraveling the mysteries of algebraic equations. Understanding how and when to use them is crucial to becoming proficient in math.

Linear Equation Evaluation

To evaluate a linear equation means to find the value of the variable that makes the equation true. In our exercise \(y=-9x-14\), the equation outlines a linear relationship between 'x' and 'y'. This implies that for every value of 'x', there is a corresponding value of 'y' that satisfies the equation.

The step by step solution shows how after substituting and performing the necessary arithmetic operations, we determined that \(y=13\) when \(x=-3\). This evaluation gives us a concrete point on the line represented by the equation, offering a snapshot of the relationship between 'x' and 'y'. In a broader context, when dealing with linear functions, we often evaluate them across a range of inputs to understand the complete 'picture' they present. In the case of our example, plotting several points where 'x' takes different values can provide us with a visual representation of a straight line, reinforcing the concept that linear equations produce linear graphs.