Problem 21 - Precalculus with Limits

Question

Solving a System of Linear Equations In Exercises \(19 - 24\) , solve the system of linear equations and check any solutions algebraically. $$\left\\{ \begin{aligned} 3 x - y & = 9 \\ x - 2 y & = - 2 \end{aligned} \right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 4\) and \(y = 3\).

Step 1: Solving first equation for x

You should first isolate \(x\) on one side of the equation in the first equation. Doing so will allow for substitution into the second equation. So, \(x\) will be expressed in terms of \(y\) as follows: \(3x = 9 + y \) , then \(x = (9 + y) / 3\).

Step 2: Substituting for x in the second equation and solving

Substitute \(x = (9 + y) / 3\) into the second equation: \((9 + y) / 3 - 2y = -2\). Multiply through by 3 to get rid of the fraction: \(9 + y - 6y = -6\). This simplifies to \( -5y = -6 - 9\). Solving for \(y\) gives \(y = 3\).

Step 3: Substituting y into the first equation and solve

Substitute \(y = 3\) into the first equation: \(3x - 3 = 9\). Which simplifies to \(3x = 12\). Solving for \(x\) gives \(x = 4\).

Step 4: Checking the solution

Finally, it's important to check the solution by substituting \(x = 4\) and \(y = 3\) into both original equations. Both simplify to true statements, confirming the solution is correct.

Key Concepts

Solving Linear Equations

A system of linear equations consists of two or more equations that share the same set of variables. In our example, we have two equations that need to be solved simultaneously. Solving such systems involves finding values for the variables that satisfy all the given equations at once.

To solve the given system, we start by dealing with one equation to express one variable in terms of the other, making the solution more manageable. Essentially, you're trying to pinpoint where the equations "intersect" or match in values.

• Identify variables within each equation. Here, these are \(x\) and \(y\).
• Find the exact values of \(x\) and \(y\) that satisfy both equations.
• Substituate one variable with a known expression from the other equation.

This process reveals the solutions or the "intersections," which are the secret to unraveling these mathematical mysteries.

Substitution Method

The substitution method is a key technique used to tackle systems of equations, particularly effective when you can easily isolate a variable. Here's how it works:

Start by choosing one equation and solving it for one variable. We'll refer back to this with our original problem:

• Take the first equation, \(3x - y = 9\), and solve for \(x\).
• This gives \(x = \frac{9 + y}{3}\).

Once you have one variable isolated, substitute that expression into the other equation. This means replacing the variable you've just expressed in terms of the other with the equivalent value.

After substitution, simplify the equation until only one variable remains, making it easy to solve. Plugging this solution back into either original equation lets you solve for the other variable, thus completing the solution.

Algebraic Solution

Algebraic solutions are the culmination of your efforts when solving equations. First, let's see what this means in practice.

After substituting and simplifying, you'll often get an equation in a more accessible form, such as \(-5y = -15\). Immediately, this allows you to directly solve for \(y\), finding that \(y = 3\).

From here, you substitute back to get \(x\). Placing \(y = 3\) into \(3x - y = 9\) returns \(x = 4\).

• Rechecking is crucial. Substitute \(x = 4\) and \(y = 3\) back into the original system:
• For \(3x - y = 9\), we get \(9 - 3 = 6\), which simplifies to \(6 = 6\) – a true statement.
• For \(x - 2y = -2\), substituting gives \(4 - 6 = -2\), also true.

This confirms our answer, revealing the algebraic solution was correctly achieved.