Problem 44 - Algebra 1: Concepts and Skills

Question

Solve the equation by simplifying first. $$ 6=y-(-11) $$

Step-by-Step Solution

Answer

The solution to the equation is \(y=-5\).

Step 1: Simplify the expression

The equation is given as \(6=y-(-11)\). Simplify the double negative, which turns into a positive, thereby the equation becomes \(6=y+11\).

Step 2: Isolate the variable

To isolate y, subtract 11 from both sides of the equation. The equation then reads \(6 - 11 = y+11 - 11\). This simplifies to \(-5=y\).

Key Concepts

Solving Equations

Solving equations involves finding the value of a variable that makes the equation true. In an equation, both sides are expressions that are equal to each other. To solve an equation like \(6 = y + 11 \), you need to perform operations that will isolate the variable, in this case, \(y\). Begin by identifying operations that can `undo` the effect of other operations. This process often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

Here’s a basic strategy for solving linear equations:

• Look at both sides of the equation to understand what operations are being performed on the variable.
• Use inverse operations to cancel out these operations, working to get the variable alone on one side.
• Perform the same operation on both sides to maintain equality.
• Simplify each step as you go until the variable is isolated.

Understanding how to balance an equation is crucial, as every operation done to one side must be done to the other to keep the equation balanced.

Simplifying Expressions

Simplifying expressions is an important step in solving equations. It involves rewriting the expression in a simpler or more efficient form. When you simplify, you're aiming to make calculations easier and more straightforward.

Key points to keep in mind when simplifying expressions include:

• Combining like terms: Terms that have the same variables raised to the same power can be combined by adding or subtracting their coefficients.
• Removing unnecessary terms: Simplify expressions to eliminate redundant terms like double negatives.
• Applying arithmetic rules: Use basic arithmetic rules, such as the distributive property, to simplify complex expressions.

In the original problem, simplifying \( y - (-11) \) involves converting the double negative into a positive, resulting in \( y + 11 \). This step makes the equation much easier to solve.

Variables in Equations

Variables are symbols used to represent unknown or changeable values in equations. They allow us to generalize mathematical expressions and solve a variety of problems.

Here’s what to remember about variables in equations:

• Variables like \(y\) in our example can take on any value that satisfies the equation.
• They are placeholders for numbers we aim to determine through solving the equation.
• Understanding the role of variables is essential for solving equations because they show what needs to be isolated or simplified.
• In solving equations, our goal often revolves around making the variable the subject of the equation.

Once the variable is isolated, interpreting its value in the context of the problem is the next step. This establishes the solution as the value assigned to the variable that makes the equation true.