Problem 39
Question
Solve the equation. $$ 34+x=10 $$
Step-by-Step Solution
VerifiedKey Concepts
Isolate the Variable
The cornerstone of solving linear equations is to isolate the variable. 'Isolating the variable' means rearranging the equation so the variable we are solving for is on one side of the equation, and all other terms are on the opposite side. For the example 34 + x = 10, this entails moving 34 to the right side of the equation to free the variable x from any other terms. We do this by performing the inverse operation: in this case, subtraction. By subtracting 34 from both sides, we follow the golden rule of keeping the equation balanced while isolating x.
Why Isolate?
Isolating the variable provides a clear view of what the variable represents in terms of the values given in the equation. Once isolated, the variable's value is revealed after simplification, allowing for a straightforward solution.
Algebraic Equations
Algebraic equations are mathematical statements that show the relationship between different variables and constants. In the context of our example, 34 + x = 10 is an algebraic equation where x is the variable, and 34 and 10 are constants. The equals sign (=) signals that the values on both sides of it are equivalent.
Equations Tell a Story
Think of an equation as a sentence that tells a numerical story. The goal is to decode this sentence to understand what number the variable represents. Algebraic equations can be simple or complex, but solving them generally involves the same principles: combining like terms, performing inverse operations, and manipulating the equation to clarify the relationship between elements.
Balancing Equations
When we manipulate an equation to solve for a variable, it is crucial to maintain balance. This means whatever is done to one side must be done to the other. This balance is what makes mathematics fair and consistent. If we subtract 34 from the left side of our example, we must also subtract 34 from the right side to preserve the equality.
The Scales Analogy
Imagine an old-fashioned scale with weights on both sides. To keep the scale balanced, any weight taken off one side must be removed from the other side as well. Applying this analogy to algebra keeps equations true while we work to isolate the variable.
Simplifying Expressions
Once the manipulation of the equation has resulted in the variable being on one side, it's time to simplify. Simplifying an expression means to perform all possible basic arithmetic in order to write the expression in its most basic form. For the initial problem, x = 10 - 34 is simplified by carrying out the subtraction, yielding x = -24.
The Final Step
The purpose of simplifying is to clearly present the answer. After simplification, the solution to the equation should be apparent, leaving no further operations necessary to find the value of the variable.