Problem 71 - College Algebra

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. An investor who dabbles in real estate invested 1.1 million dollars into two land investments. On the first investment, Swan Peak, her return was a 110\(\%\) increase on the money she invested. On the second investment, Riverside Community, she earned 50\(\%\) over what she invested. If she earned \(\$ 1\) million in profits, how much did she invest in each of the land deals?

Step-by-Step Solution

Answer

The investor invested $750,000 in Swan Peak and $350,000 in Riverside Community.

Step 1: Define Variables

Let \( x \) be the amount invested in Swan Peak and \( y \) be the amount invested in Riverside Community.

Step 2: Set Up Equations Based on the Total Investment

The sum of the investments in Swan Peak and Riverside Community is 1.1 million dollars. This can be expressed as the equation: \[ x + y = 1,100,000. \]

Step 3: Set Up Equation Based on Profit

Profit from Swan Peak is \( 110\% \) of \( x \) and profit from Riverside Community is \( 50\% \) of \( y \). The total profit is 1 million dollars, leading to the equation: \[ 1.1x + 0.5y = 1,000,000. \]

Step 4: Solve the System of Equations

We have a system of equations: 1) \( x + y = 1,100,000 \) and 2) \( 1.1x + 0.5y = 1,000,000 \). Solve this system using substitution or elimination. We will use substitution: solve the first equation for \( y \): \[ y = 1,100,000 - x. \] Substitute this expression for \( y \) into the second equation: \[ 1.1x + 0.5(1,100,000 - x) = 1,000,000. \]

Step 5: Simplify and Solve for \( x \)

Simplify the equation by distributing \( 0.5 \): \[ 1.1x + 550,000 - 0.5x = 1,000,000. \] Combine like terms: \[ 0.6x + 550,000 = 1,000,000. \] Subtract 550,000 from both sides: \[ 0.6x = 450,000. \] Divide both sides by 0.6 to solve for \( x \): \[ x = 750,000. \]

Step 6: Solve for \( y \)

Substitute \( x = 750,000 \) back into the equation \( y = 1,100,000 - x \): \[ y = 1,100,000 - 750,000 = 350,000. \]

Step 7: Verify the Solution

Check that \( x = 750,000 \) and \( y = 350,000 \) satisfy both original equations. For \( x + y = 1,100,000 \), \( 750,000 + 350,000 = 1,100,000 \), which is correct. For \( 1.1x + 0.5y = 1,000,000 \), \( 1.1(750,000) + 0.5(350,000) = 825,000 + 175,000 = 1,000,000 \), which is also correct.

Key Concepts

Linear Equations

Linear equations are essential in solving many real-world problems, especially those involving relationships between two quantities. A linear equation typically involves variables raised to the power of one and constants.
In our investment problem, we use two linear equations to represent the different conditions provided. One equation accounts for the total amount invested, and the other represents the total profit.
These equations are straightforward and manageable because they lead to straight-line graphs when plotted on a coordinate plane. Establishing these equations with appropriate variables is a vital first step in finding the solution. This approach demonstrates the power of algebra to model and solve practical problems.

Investment Problems

Investment problems often require analyzing how funds are distributed across different ventures. In this exercise, the investor has allocated a total of $1.1 million into two real estate investments.
Investment problems usually involve determining the allocation of resources to maximize returns or meet specific goals within given constraints.

• The sum of investments is crucial here, meaning the combined capital for both ventures totals $1.1 million.
• Each investment produces different returns, represented by different profit percentages.

Understanding the concepts behind these calculations helps unpack complex scenarios found in personal finance and business.

Solving Algebraic Equations

Solving algebraic equations is a fundamental skill in mathematics. Here, it involves finding the values of unknown variables that satisfy all equations simultaneously.
To solve a system of equations, techniques like substitution or elimination are employed. The substitution method used here involves:

• Expressing one variable in terms of the other in one equation.
• Substituting it in the second equation to find the second variable.

Breaking down the algebraic process into simpler steps allows for rigorous verification and minimizes errors, leading to the correct solution.

Profit Calculation

Profit calculation is an important aspect of financial literacy. It gives insights into how much gain is achieved compared to the initial investment.
For the properties in this problem, the profit percentages are used to form equations that capture the financial outcome. Swan Peak yields a 110% return on investment, while Riverside Community brings a 50% return.

• The total gain from both investments must meet the specified profit condition of $1 million.
• The combination of these elements results in a precise calculation of financial success or failure.

Accurate profit calculation provides better planning and assessment of investment potential.