Problem 46 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value of \(x\) is 7.

Step 1: Understand the Equation

The given equation is \(x + y = 5\). It means that the sum of \(x\) and \(y\) equals 5.

Step 2: Substitute the value of \(y\)

We know \(y = -2\). We can substitute this value into the equation \(x + y = 5\) to get \(x - 2 = 5\).

Step 3: Solve for \(x\)

Add 2 to both sides of the equation: \(x - 2 + 2 = 5 + 2\). This simplifies to \(x = 7\).

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations by substituting one equation into another. It is especially effective when one of the variables in a system is already isolated or can be easily isolated. In the context of prealgebra, this method helps us solve for an unknown variable by substituting the value of another variable, if it's already known. For example, in our problem, we know that the value of the variable \( y \) is given as \(-2\).
To use the substitution method, follow these simple steps:

• Isolate one variable (if not already provided). In this case, it's already done since we have the value for \( y \).


• Substitute the known value into the other equation. Here, we substitute \( y = -2 \) into \( x + y = 5 \).


• Solve the new equation to find the unknown variable. For this, we end up with \( x - 2 = 5 \).

By substituting, you turn the original problem into a simpler arithmetic puzzle. This is why the substitution method is favored when certain values are already clearly defined.

Solving Linear Equations

Solving linear equations is a fundamental skill in mathematics, particularly in prealgebra. A linear equation is an equation that forms a straight line when graphed, typically written in the form \(ax + by = c\) or simplified to one variable. The goal of solving a linear equation is to find the value of the unknown variable that makes the equation true.
In our exercise, we started with the equation \(x + y = 5\). After substituting \(y = -2\), the equation became \(x - 2 = 5\). Solving this involves:

• Isolating the variable on one side of the equation. We do this by performing the same operation on both sides to maintain balance.


• In this case, adding 2 to both sides simplifies the equation to \(x = 7\).

Remember, whatever operation you perform on one side, you must also perform on the other side to keep the equation balanced, like a scale.

Integers

Integers are a set of numbers that include all whole numbers and their negative counterparts, like \(..., -3, -2, -1, 0, 1, 2, 3, ...\). Integers can be positive, negative, or zero, and do not include fractions or decimals.
In our problem, \(y\) is given as \(-2\), which is a negative integer. When solving equations with integers, it's important to pay attention to the sign of each number.
Here's a quick tip when working with integers:

• When you subtract a negative number, it's like adding its positive counterpart. For instance, \(-2\) becomes \(+2\) when subtracted.


• Adding or subtracting integers with different signs will depend on which absolute value is greater, and the final answer will take the sign of the larger absolute value.

By understanding and using the properties of integers, you can confidently solve linear equations and other mathematical problems involving these numbers.