Problem 46 - Algebra 1

Question

(3) GARDEN PLANTS In early spring, you buy 6 potted tomato plants for your container garden. The plants contained in 8-inch pots sell for \(\$ 5\) and the plants contained in 10 -inch pots sell for \(\$ 8 .\) If you spend \(\$ 36,\) how many of each size are you buying? [Number of 8 -inch pots \(+[\text { Number of } 10 \text { -inch pots }]=\) Total number buying] Cost of all \(8 \text { -inch pots }+\text { Cost of all } 10 \text { -inch pots }]=\) Total cost]

Step-by-Step Solution

Answer

If the solution to the system of equations is \(x = 4\), \(y = 2\), then 4 pots of 8-inch size and 2 pots of 10-inch size were bought.

Step 1: Formulate the Two Equations

Let \(x\) be the number of 8-inch pots and \(y\) be the number of 10-inch pots. Then, the two equations based on the provided information are: \[x + y = 6\] (total number of pots) \[5x + 8y = 36\] (total cost)

Step 2: Solve the System of Equations

This system of equations can be solved either by substitution or by elimination. Let's use substitution for this example. Solve the first equation for \(x\), to get \(x = 6 - y\). Substitute \(x = 6 - y\) into the second equation: \[5*(6 - y) + 8y = 36\]. Solve for \(y\).

Step 3: Substitute value of \(y\) in equation 1

Once the value of \(y\) is obtained, it can be substituted in the first equation to determine \(x\).

Step 4: Verify the Solution

Substitute the values for \(x\) and \(y\) into both initial equations to verify that they are correct.

Key Concepts

Linear Equation

A linear equation is a fundamental concept in algebra that represents a straight line when plotted on a graph. It has the form ax + by = c, where a, b, and c are constants, and x and y are variables. These equations show relationships between two quantities that change at a constant rate. For instance, in our exercise, the cost of the tomato plants and their quantity are directly related.

In real-life situations like buying potted plants, linear equations can represent different aspects such as cost, quantity, or measurements. By formulating such equations from a word problem, the scenario is translated into a mathematical format that can be easily worked with. This simplification is the first step in solving many algebraic problems and is crucial when dealing with systems of equations.

Substitution Method

The substitution method is a way to solve a system of equations where one equation is solved for one variable, which is then plugged into the other equation. It is particularly useful when equations are not easily aligned for elimination or when one equation is already solved for a single variable.

In our garden plants problem, we solve the first equation for x to find an expression that can substitute x (in terms of y), which is then used in the second equation. This reduction of the original system to a single equation in one variable allows for a straightforward solution. y can be isolated, and its value is found.

Substituting this value back into the original expression for x gives us the quantity of each pot size bought. This methodical approach streamlines the problem-solving process and is excellent for beginners who are learning how to manipulate algebraic expressions.

Algebraic Problem Solving

Algebraic problem solving is a methodical process that translates real-world situations into mathematical form, which then can be manipulated to find a solution. It involves understanding the problem, representing it with equations or expressions, performing algebraic operations, and checking the solutions for accuracy.

In the context of our potted plants example, the algebraic problem-solving process begins with formulating equations based on the given scenario. Calculations are performed using methods like the substitution method, as demonstrated. Once a solution is reached, it's important to verify the results by substituting the numbers back into the original equations, ensuring they hold true. This not only confirms the answer but also solidifies the learner's understanding of the relationships between variables within the system of equations.

Enhancing students' problem-solving skills with algebra is essential, as it promotes logical thinking, precision, and the ability to apply mathematical concepts to various fields beyond academia.