Problem 31 - Algebra 1: Concepts and Skills

Question

You designate one row in your garden to broccoli and pea plants. Each broccoli plant needs 12 inches of space and each pea plant needs 6 inches of space. The row is 10 feet (120 inches) long. If you want a total of 13 plants, how many of each plant can you have?

Step-by-Step Solution

Answer

There can be 8 broccoli and 5 pea plants.

Step 1: Identify the variables and the constraints

Let's say B represents the number of broccoli plants and P represents the number of pea plants. From the problem, we know that the total length of the row is 120 inches and each broccoli and pea plant requires 12 and 6 inches respectively. This gives us the first equation: 12B + 6P = 120. Also, it is stated that the total number of plants is 13. Thus, the second equation is B + P = 13.

Step 2: Transform one equation and substitute

From the second equation B = 13 - P, we then substitute this into the first equation replacing B, which gives: 12(13 - P) + 6P = 120.

Step 3: Solve for P

Solving the equation 12(13 - P) + 6P = 120, we get P = 5. This means there are 5 pea plants.

Step 4: Substitute P into the second equation to find B

Substitute P = 5 into the second equation B = 13 - P, we get B = 8. This means that there are 8 broccoli plants.

Step 5: Validate the solution

Check if these solution values hold true in the original equations. For 12B + 6P = 120, substituting B=8, P=5 gives 96 + 30 = 120 which is correct. Also for B + P = 13, 8 + 5 = 13 which is also correct. Thus the solution is valid.

Key Concepts

Algebraic Problem-Solving

Algebraic problem-solving is like unraveling a mystery using mathematical clues. It involves understanding the problem, identifying the unknowns, and applying algebraic methods to find the solution.

In our garden row example, the mystery is figuring out the number of broccoli and pea plants we can fit. The clues are the space required by each plant and the total space available. The unknowns are the number of broccoli (B) and pea plants (P).

The first step is translating the word problem into mathematical equations using the clues given. We established two equations to represent the constraints: the space each plant needs and the total number of plants we aim for. These equations form a system that we can solve using various algebraic methods.

One crucial aspect of algebraic problem-solving is ensuring your solution makes sense within the context of the problem. After finding a numerical answer, it's important to verify if the numbers are logically consistent with the original situation, rejecting solutions that don't fit the real-world constraints.

Linear Equation Substitution Method

The linear equation substitution method is a technique to solve systems of equations where one variable is expressed in terms of the other and then substituted into another equation. This method is particularly useful when dealing with two-variable systems.

In our scenario, after formulating our equations, we transformed one of them to express B in terms of P. This rearrangement allowed us to replace B with its equivalent expression in the other equation, reducing the system to a single equation with one variable. In this compressed form, it becomes much simpler to solve for the single variable.

Once we found the value for P, we used it to determine B. Substitution is both powerful and elegant; it takes the complexity of a system and breaks it down into more manageable single-variable equations. Hence, learners find it rewarding as it offers a clear path from problem statement to solution.

Variable Representation in Algebra

In algebra, variable representation is the practice of using symbols, typically letters, to stand in for unknown or variable quantities. It's like casting actors to play roles in a script; each variable has a part to play in the mathematical narrative we're constructing.

Our garden planning problem uses B and P as variables representing the quantities of broccoli and pea plants. Variables allow us to abstract and generalize problems, so we can focus on the relationships and operations holding them together. This abstraction is why you can use algebra to model an incredible array of situations.

Choosing appropriate variable symbols is part of clear algebraic representation. It's why we often use the first letter of the items we're counting - it keeps our mathematical story straight in our minds. As variables hold places for numbers, they must be treated according to mathematical rules, leading to correct and meaningful solutions.