Problem 47 - Algebra 1

Question

Solve the equation. $$ 3 x+7=-2 $$

Step-by-Step Solution

Answer

The solution to the equation \(3x + 7 = -2\) is \(x = -3\).

Step 1: Subtract 7 from both sides

To begin solving for x, the equation needs to be rewritten in a way such that numbers and variables are isolated on different sides of the equal symbol. This can be achieved by performing the same operation on both sides of the equation to maintain equality. In this case, 7 could be subtracted from both sides, resulting in the equation \(3x = -9\)

Step 2: Divide both sides by 3

The next step is to isolate x on one side of the equation. This is done by dividing both sides by 3 because x is currently being multiplied by 3 in the equation \(3x = -9\). After division by 3, the equation becomes \(x = -3\)

Step 3: Check the solution

After finding the solution x = -3, confirmation is needed to ensure it is correct. This is done by substituting x = -3 in the initial equation. Therefore, \(3(-3) + 7 = -2\), which simplifies to -9 + 7 = -2. This confirms the found solution is correct.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They often involve variables, coefficients, and constants. In the context of the original exercise \(3x + 7 = -2\), the equation represents a balance that must be maintained. A fundamental principle when solving algebraic equations is that whatever operation you perform on one side, you must also perform on the other to maintain that balance.

To effectively solve algebraic equations, it's essential to follow a structured process, which typically involves simplifying expressions and combining like terms, if necessary. Understanding the properties of equality and being familiar with operations on integers and rational numbers are foundational skills that aid in solving such equations.

Isolating Variables

Isolating variables is a critical step in solving algebraic equations. To isolate a variable means to get the variable by itself on one side of the equation, with a coefficient of 1. The goal is to have this form: \(x = \text{some value}\).

The strategy to isolate variables can involve several operations, including adding or subtracting terms on both sides of the equation to remove constants or dividing or multiplying both sides by a nonzero coefficient to make the coefficient of the variable equal to 1. It's necessary to perform these operations with precision to avoid errors. For example, \(3x = -9\) can be transformed to \(x = -3\) by dividing both sides of the equation by 3, which effectively isolates \(x\).

Equation Operations

Equation operations are the mathematical procedures used on an equation to solve for a variable. These can include addition, subtraction, multiplication, and division. We also consider operations like factoring, distributing, and combining like terms as part of equation operations.

When performing these operations, it's essential to keep the equation balanced by doing the same to both sides as demonstrated in the example problem. Subtracting 7 from both sides changed the equation from \(3x + 7 = -2\) to \(3x = -9\), and dividing by 3 then isolated \(x\). It's also important to note that some operations will change the form but not the value of expressions, while others can simplify the equation to make it easier to solve.

Checking Solutions

Checking solutions is the final, but no less critical, step in the process of solving algebraic equations. After a potential solution is found, it must be validated to ensure it indeed makes the original equation true.

To check solutions, the found value of the variable is substituted back into the original equation. If both sides of the equation remain equal after substitution, then the solution is correct. For instance, substituting \(x = -3\) back into the original equation \(3x + 7 = -2\) gives us \(3(-3) + 7 = -9 + 7 = -2\), which verifies the solution is accurate. This step is vital because it confirms the integrity of the solution process and provides assurance that the answer is indeed a solution to the equation.