Problem 63 - Intermediate Algebra

Question

Solve each system of equations. $$ \left\\{\begin{array}{l} x=\frac{2}{3} y \\ y=4 x+5 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = -2\) and \(y = -3\).

Step 1: Substitute First Equation

Substitute the expression for \(x\) from the first equation \(x = \frac{2}{3} y\) into the second equation. This gives us \( y = 4 \left( \frac{2}{3} y \right) + 5 \).

Step 2: Simplify the Equation

Simplify the substituted equation by calculating \( 4 \cdot \frac{2}{3}\). This results in \( \frac{8}{3} y \) so the equation becomes \( y = \frac{8}{3} y + 5 \).

Step 3: Solve for y

Bring all terms involving \( y \) to one side of the equation by subtracting \( \frac{8}{3} y \) from both sides. This leads to \( y - \frac{8}{3} y = 5 \). Simplifying gives \( -\frac{5}{3} y = 5 \). Now solve for \( y \) by multiplying both sides by \(-\frac{3}{5}\), resulting in \( y = -3 \).

Step 4: Solve for x

Use the value of \( y = -3 \) in the first equation \( x = \frac{2}{3} y \) to find \( x \). Substitute \( y = -3 \) to obtain \( x = \frac{2}{3} \times (-3) = -2 \).

Step 5: Verify the Solution

Substitute \( x = -2 \) and \( y = -3 \) into the second equation to ensure the solution satisfies it. The second equation is \( y = 4x + 5 \). Substitute to check: \( -3 = 4(-2) + 5 \) simplifies to \( -3 = -8 + 5 \), which confirms that \( -3 = -3 \).

Key Concepts

Substitution Method

The substitution method is a strategy to solve a system of equations by expressing one variable in terms of another. It's a logical approach that allows you to simplify the complexity of dealing with two variables simultaneously. Here's how it works:

• Start with a system of equations, where one of the equations is solved for one variable. In our example, the first equation is already solved for \( x \) as \( x = \frac{2}{3} y \).


• Substitute this expression for the variable into the second equation. This step effectively reduces the problem from two variables to a single variable, making it more straightforward to solve. By replacing \( x \) in the second equation with \( \frac{2}{3} y \), we simplify the problem.


• Solve for the remaining variable. Once substituted, the equation's complexity reduces dramatically, allowing you to find the precise value of the variable.

The substitution method is particularly helpful when one equation is easily solvable for one variable. It can turn a seemingly complex problem into a manageable one, allowing you to systematically arrive at a solution.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. This is often a straightforward process, especially when dealing with one variable:

• First, simplify both sides of the equation as much as possible, using fundamental arithmetic operations.


• Next, isolate the variable by moving all terms involving the variable to one side of the equation. This may involve adding, subtracting, multiplying, or dividing.


• Once isolated, solve for the variable by performing the inverse operation of what's been done to it.

In our example, once we substituted to find the equation \( y = \frac{8}{3} y + 5 \), we simplified by bringing terms together and solving \( -\frac{5}{3} y = 5 \). Solving this gave us the precise value of \( y \). Linear equations are direct and usually result in one distinct answer.

Verification in Mathematics

Verification in mathematics is a crucial step that ensures the solutions obtained from calculations actually satisfy the original equations. It's like a safety check to confirm the accuracy of your work:

• After obtaining potential solutions, substitute these values back into the original equations to see if they hold true. If they do, you've likely found correct solutions.


• In practice, as seen in our problem, once you solve for \( x \) and \( y \), plug these values back into one of the original equations. For example, replacing \( x = -2 \) and \( y = -3 \) into the second equation \( y = 4x + 5 \) confirmed our solution was accurate, as both sides equaled -3.


• This verification step is essential because it helps you catch errors in processing or calculation.

Verifying your answers can save time and effort by ensuring correctness right away, avoiding the need to redo work down the line.