Problem 49 - Prealgebra

Question

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

Step-by-Step Solution

Answer

If \(y = -3\), then \(x = 8\).

Step 1: Identify the known values

We know that the equation given is \(x + y = 5\) and we are given \(y = -3\).

Step 2: Substitute the known value into the equation

Replace \(y\) in the equation \(x + y = 5\) with \(-3\). The equation now becomes \(x - 3 = 5\).

Step 3: Solve for \(x\)

To isolate \(x\), add 3 to both sides of the equation: \(x = 5 + 3\).

Step 4: Compute the result

Calculate the right-hand side: \(5 + 3 = 8\). Therefore, \(x = 8\).

Key Concepts

Addition in Solving Linear Equations

Addition is a fundamental operation in mathematics and particularly valuable when solving linear equations like the one in this exercise. In the context of linear equations, addition can help isolate variables to find their values. When we have an equation in the form of \(x + y = 5\), and we need to solve for one of the variables, we use addition to simplify the equation.
For instance, if we're given \(x - 3 = 5\), we can add 3 to both sides of the equation. This eliminates the -3 next to \(x\), leaving us with \(x = 8\).

A few key points about using addition in linear equations:

• It maintains the equality of both sides of the equation.
• It is often used to "cancel out" numbers on one side of the equation, simplifying it.
• It's a reversible operation, meaning you can undo it if necessary by subtraction.

Understanding and applying addition correctly allows you to solve equations fast and accurately.

Substitution Method

Substitution is a powerful technique used to solve equations, especially when dealing with systems of equations or specific values. This involves replacing variables with known values. In essence, you 'substitute' or 'swap' the variable with a value that has been provided or previously calculated.
In the original exercise, we had \(x + y = 5\) and were given \(y = -3\). By substituting \(y\) with \(-3\), the equation changed to \(x - 3 = 5\). This substitution is crucial because it translates the equation into a simpler form that is easier to solve.

Why substitution is useful:

• It reduces complex equations into simpler ones.
• It helps in breaking down multi-step problems into manageable steps.
• It allows you to focus on one unknown at a time, which is particularly helpful in exercises with multiple variables.

Mastering substitution can significantly ease the process of solving equations, making it an essential skill for students.

Steps in Solving Equations

Solving equations is a core skill in mathematics and involves finding the value of unknown variables that satisfy given equations. The original exercise is a clear example of solving an equation. Here's a brief rundown of the typical steps involved:
1. **Understand the Equation**: Identify what's given and what needs to be found. This involves looking at the equation and recognizing known values, just like we had \(x + y = 5\) and needed to solve for \(x\).
2. **Substitute Known Values**: Replace variables with their known values, such as substituting \(y = -3\) in our equation.
3. **Isolate the Variable**: Use arithmetic operations like addition, subtraction, multiplication, or division to get the unknown variable on one side of the equation. With our exercise, adding 3 to both sides cleared up the equation to solve for \(x\).
4. **Calculate the Result**: Perform the final computation to find the value of the unknown. In the case of our exercise, adding up \(5 + 3\) resulted in \(x = 8\).

Following these steps systematically enables students to tackle linear equations confidently. Consistent practice with these steps is key to becoming proficient in mathematics.