Problem 19 - Algebra 1: Concepts and Skills

Question

Choose a solution method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{array}{c} {2 x-y=3} \\ {4 x+3 y=21} \end{array} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 3\) and \(y = 3\).

Step 1: Multiply the Equations

Multiply the first equation by 3 and the second by 1 to have the same coefficients for y in both equations, to allow for their elimination. By doing so, the linear system then becomes: \[ \begin{array}{c} {6 x-3 y=9} \ {4 x+3 y=21} \end{array} \]

Step 2: Add the Equations

Now, add these two equations together. Doing so, the term \( -3y \) and \( +3y \) will cancel out each other and result in: \[10x = 30\]

Step 3: Solve for x

Now, solve for x by dividing both sides of the equation by 10 to get: \[x = 3\]

Step 4: Substitute x in the Original Equation

Next, substitute the value of x in one of the original equations. Substituting x = 3 in the first equation \(2 x - y = 3\) gives: \[2 \cdot 3 - y = 3\]

Step 5: Solve for y

Then, solve for y by reordering the equation to get: \[y = 2 \cdot 3 - 3 = 3\]

Key Concepts

Solution Methods

When tackling a system of linear equations, choosing an effective solution method is crucial. There are several common methods to choose from, each with its own advantages. Here are the most popular ones:

• Substitution: Best used when one equation is already solved for one variable.
• Elimination: Ideal when you can easily eliminate one variable by adding or subtracting the equations.
• Graphing: Useful for visualizing the intersection point of two equations, suitable when precise calculations are less critical.

In the solution provided, the elimination method was chosen because it allowed for straightforward removal of variable \( y \) by adjusting the coefficients. This is particularly helpful when the coefficients of \( x \) or \( y \) are easily manipulable to match each other, facilitating the elimination.

Substitution

The substitution method is a staple in solving systems of equations. This method involves expressing one variable in terms of another and then substituting this expression into the other equation. Here’s how it works:

• Choose an equation and solve it for one of the variables.
• Replace this variable in the second equation with the expression found.
• Solve the resulting equation for the remaining variable.

Using the substitution method can be really handy when one of the equations is simple, making isolating a variable a breeze. However, in this exercise, elimination was a more direct approach, but substitution was still used in Step 4 to find \( y \) after \( x \) was determined.

Elimination

Elimination, also known as the "addition/subtraction" method, is a powerful tool for solving systems of equations. The aim is to eliminate one of the variables by aligning and adding/subtracting the equations. Here's the general idea:

• Multiply equations (if necessary) to get equal coefficients for one of the variables.
• Add or subtract the equations to cancel out one variable.
• Solve for the remaining variable.

In the problem solved, the equations were manipulated so that \( -3y \) and \( +3y \) canceled each other out, quickly leading to the calculation of \( x = 3 \). The elimination method was particularly suitable here because of the straightforward cancellation of \( y \).

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. In this exercise, we have the system:
\[\begin{array}{c} {2 x-y=3} \ {4 x+3 y=21} \end{array}\]
There are typically three outcomes for systems:

• One Solution: The system intersects at precisely one point, as in this exercise where \( x = 3 \) and \( y = 3 \).
• No Solution: The system has parallel lines and no intersection points (inconsistent).
• Infinite Solutions: The system has dependent equations, so they overlap completely.

Understanding these outcomes helps decide the best method to solve the system and foresee the nature of the solution.