Problem 44 - Prealgebra

Question

Suppose \(y=3 x-2 .\) Find \(y\) if: $$x=5$$

Step-by-Step Solution

Answer

When \(x = 5\), \(y = 13\).

Step 1: Identify the Given Equation

The problem provides the equation relating two variables, which is \(y = 3x - 2\). In this equation, \(y\) is represented as a linear function of \(x\).

Step 2: Substitute the Given Value

You need to find \(y\) when \(x = 5\). Substitute \(5\) into the equation wherever \(x\) appears: \(y = 3(5) - 2\).

Step 3: Perform Multiplication

Calculate the multiplication part of the equation first. Compute \(3 \times 5 = 15\). The equation now updates to \(y = 15 - 2\).

Step 4: Perform Subtraction

Subtract \(2\) from \(15\) to find \(y\): \(y = 15 - 2 = 13\).

Step 5: Conclusion

After solving the equation, we determine that when \(x = 5\), \(y\) is equal to 13.

Key Concepts

Solving Equations

When it comes to solving equations, it is all about finding the value of one variable in terms of another. Here, we're dealing with linear equations, which are equations of the first degree. They have the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. Linear equations are straightforward and involve basic arithmetic operations like addition, subtraction, multiplication, and division.

The primary objective while solving an equation is to isolate the variable of interest on one side of the equation. In doing so, we often perform the same operation on both sides of the equation to keep it balanced. The ultimate goal is to find the exact value of the unknown variable that satisfies the equation. In our example, we found \( y \) given \( x = 5 \) by substituting \( x \) into the given linear equation.

Substitution Method

The substitution method is an effective technique for solving equations involving two or more variables. It involves replacing a variable with its equivalent value or expression. In simpler terms, if we have two equations, we solve one of the equations for one variable and then substitute this expression back into the other equation.

Here’s how you can think of it step-by-step:

• First, solve one of your equations for one of the variables.
• Next, substitute this expression into the other equation, replacing the variable you just solved for.
• Lastly, solve the new equation for the remaining variable.

This method efficiently reduces a system of equations to a single variable equation, making it easier to find a solution. In our given problem, substitution was used when we inserted the value \( x = 5 \) into the equation \( y = 3x - 2 \), enabling us to find the corresponding \( y \) value.

Mathematics Education

In the realm of mathematics education, mastering various problem-solving techniques is key to developing strong analytical and critical thinking skills. Linear equations, such as those we explored in this exercise, are part of the foundation upon which more advanced mathematical concepts are built.

Learning to solve these equations involves distinct strategies such as the substitution method, and it also focuses on teaching students processes like simplification, isolation of variables, and logical reasoning. Building a solid understanding of these principles not only helps students solve equations but also strengthens their overall math literacy.

By understanding problems, identifying the appropriate methods, and applying them accurately, students gain confidence in mathematics. This approach in mathematics education encourages a deeper comprehension, making mathematics less intimidating and more accessible to learners of all levels.