Problem 40 - Algebra 1

Question

Solve the equation. $$-3=-a+(-4)$$

Step-by-Step Solution

Answer

The solution of the equation is \(a=1\).

Step 1: Setup the equation

The primary step is to write out the original equation. Here, the given equation is \(-3=-a+(-4)\).

Step 2: Simplify the equation

Simplify the equation by removing the brackets in the equation. The simplified equation will be: \(-3=-a-4\).

Step 3: Isolation of variable 'a'

The goal is to isolate the variable 'a' on one side of the equation. This can be done by restructuring the equation by adding 'a' and '3' to both sides of the equation. The restructured equation will be: \(a=4-3\).

Step 4: Solve for 'a'

In this final step, simply subtract 3 from 4 to solve for 'a'. Therefore, \(a=1\).

Key Concepts

Variable Isolation

Understanding how to isolate a variable is a foundational skill in algebra. Variable isolation involves manipulating an equation to get the variable by itself on one side of the equation. In the exercise \( -3=-a+(-4) \), the goal is to solve for \( a \). To isolate \( a \), you need to perform the same operation on both sides of the equation, maintaining equality. Our exercise demonstrates this by adding \( a \) to both sides, which effectively cancels it out on the right side, and then adding 3 to both sides to move the -3 over to the other side.

It may help to visualize it as balancing scales; whatever you do to one side, you must do to the other to keep it balanced. Remember, the purpose is to get the variable \( a \) by itself so that we have an expression like \( a = \text{value} \) which gives us the solution.

Equation Simplification

Simplifying an equation makes it easier to solve. This process entails combining like terms and removing unnecessary parentheses. In our given problem, the equation simplification step changes \( -3=-a+(-4) \) to \( -3=-a-4 \) by understanding that adding a negative is the same as subtracting.

Simplification is not just about making an equation shorter; it's about reducing it to a form where variable isolation becomes straightforward. Keep an eye out for opportunities to combine numbers and variables to streamline the equation. While simplification may sometimes involve several steps, such as distributing and combining like terms, the goal is to lead you to a point where the variable(s) can be easily isolated.

Algebraic Expressions

An algebraic expression is a collection of numbers, variables, and operations. The solution process involves working with these expressions to discover the value of variables. In our example, after isolating the variable \( a \) to one side, we end up with an algebraic expression \( a = 4 - 3 \), which tells us that \( a \) is equal to the result of the subtraction 4 minus 3.

This final expression is simple, yet it represents an essential concept in algebra: expressions are sentences that convey mathematical ideas. They can be manipulated following the rules of arithmetic to uncover the values hidden within them, just like we did to solve for \( a \) by performing the subtraction to find out that \( a = 1 \).