Problem 35 - Elementary Algebra

Question

Solve each of the conditional equations. $$ y-8=-1 $$

Step-by-Step Solution

Answer

Answer: The value of \(y\) in the given equation is \(7\).

Step 1: Identifying the current equation

The given equation is: $$ y-8=-1 $$

Step 2: Isolating y

We need to add 8 to both sides of the equation to isolate \(y\): $$ y-8+8=-1+8 $$

Step 3: Simplifying the equation

After adding 8 to both sides, the equation becomes: $$ y=7 $$

Step 4: Solution

The solution to the equation \(y-8=-1\) is \(y=7\).

Key Concepts

Isolation of Variables

When solving linear equations, one of the most important steps is the process of isolating the variable. This means rearranging the equation so that the variable we are trying to solve for is alone on one side of the equation. Doing this makes it easier to determine its value.In the equation given, \(y - 8 = -1\), our goal is to get \(y\) alone on one side. To achieve this, determine what has been done to the variable. In this case, 8 has been subtracted from \(y\). We'll need to do the opposite operation to remove it. Remember, isolating the variable is a critical step in any equation-solving process. It is the foundation upon which the other steps rely.
Without entirely isolating the variable, we cannot accurately determine its value.

Addition Property of Equality

The addition property of equality is a fundamental concept in algebra. It states that you can add or subtract the same value from both sides of an equation without changing the equation's equality. This is crucial when we need to modify one side of an equation to isolate a variable.In the original equation \(y - 8 = -1\), we noticed that \(y\) was not isolated because of the \(-8\) being subtracted from it. By applying the addition property of equality, we can add 8 to both sides (canceling out the \(-8\) on the left).

• Left side: \(y - 8 + 8 = y\)
• Right side: \(-1 + 8 = 7\)

After performing this step, the equation simplifies to \(y = 7\). The addition property of equality allows us to manipulate equations so we can isolate variables and find their values effectively, rendering this property a potent tool in solving equations.

Simplifying Equations

Simplifying equations is a key process in solving linear equations, as it allows us to express the equation in its simplest form. This step also confirms that the value we find for the variable satisfies the original equation.After applying the addition property of equality to our original equation \(y - 8 = -1\), we simplified it to \(y = 7\). Simplification is essentially the process of condensing the expression to its most concise form.Once simplified, we should always consider verifying the solution by substituting it back into the original equation. Substituting \(y = 7\) into the original equation:

• \(7 - 8 = -1\)

The left-hand side equals the right-hand side, confirming our solution is correct. Simplifying accurately and verifying results is integral to ensuring the solution's reliability.