Problem 15 - Algebra 1: Concepts and Skills

Question

Solve the formula for the indicated variable. Show all your steps. Then evaluate the new formula by substituting the given values. Solve \(A=\ell w\) for \(\ell\) Find the value of \(\ell\) when \(A=112\) and \(w=7\)

Step-by-Step Solution

Answer

The length (\(\ell\)) is 16 units.

Step 1: Solve for Length

To solve for length, \(\ell\), divide both sides of the equation by width, \(w\). It will yield a new formula: \(\ell = \frac{A}{w}\)

Step 2: Substitute Values

Substitute the given values, \(A=112\) and \(w=7\), into the new equation. So, it becomes \(\ell = \frac{112}{7}\)

Step 3: Simplify

Simplify the above equation to find the value of length. That gives us \(\ell = 16\)

Key Concepts

Solving Equations

In algebra, solving equations is a fundamental skill that involves finding the value of an unknown variable that makes the equation true. When given an equation with a formula such as \[ A = \ell \cdot w \]the task is often to isolate and solve for a specific variable, like \( \ell \). This requires using inverse operations to break down the equation step by step.

If you're beginning with the equation, first identify what operations are connecting your variables. By seeing multiplication in the term \( \ell \cdot w \), you know that the opposite operation, division, will help you isolate one term from the multiplication. Thus, dividing both sides by \( w \) gives you \( \ell \) on its own, as shown:\[ \ell = \frac{A}{w} \]

Always ensure to perform the same operation on both sides of the equation. This keeps the equation balanced and steps you closer to the solution.

Variable Manipulation

Variable manipulation refers to the operations we perform on variables in order to re-arrange equations or solve them. With an equation like \[ A = \ell \cdot w \],rearranging involves manipulating the variables to solve for \( \ell \). Start by moving terms around using basic algebra operations, such as addition, subtraction, multiplication, and division.

The goal is to express the equation in terms of the desired variable. In this case, you divided the entire equation by \( w \), the coefficient of \( \ell \), to isolate \( \ell \). This process of re-arranging simplifies to the formula:\[ \ell = \frac{A}{w} \]

Such manipulation is a crucial algebra skill, allowing you to rearrange and express equations in more useful or desired forms. This skill makes more complex problems manageable and easier to solve.

Substitution Method

The substitution method is a powerful tool in algebra where specific values are substituted into an equation to find the unknown variable's value. This method often follows re-arranging an equation to isolate a variable.

Take the newly formed equation:\[ \ell = \frac{A}{w} \]

if you know that \( A = 112 \) and \( w = 7 \), substitute these values in place of \( A \) and \( w \) respectively:\[ \ell = \frac{112}{7} \]

After substitution, compute the value to solve for \( \ell \). In this example, dividing 112 by 7 gives:\[ \ell = 16 \]

The substitution method allows you to evaluate and find numerical solutions easily once the equation is structured correctly.