Problem 29 - A Graphical Approach to College Algebra

Question

Solve each system by elimination. $$\begin{array}{l}3 x-y=-4 \\\x+3 y=12\end{array}$$

Step-by-Step Solution

Answer

The solution is \((0, 4)\).

Step 1: Write the System of Equations

We are given the following system of equations: \[\begin{align*}3x - y &= -4 \x + 3y &= 12\end{align*}\] We will solve these equations using the elimination method.

Step 2: Align for Elimination

The elimination method requires us to eliminate one of the variables by adding or subtracting the equations. Let's aim to eliminate \(y\). We have \(-y\) in the first equation and \(3y\) in the second. Multiply the first equation by 3 to align the \(y\) terms for elimination:\[\begin{align*}3(3x - y) &= 3(-4)\end{align*}\]Resulting in:\[\begin{align*}9x - 3y &= -12\end{align*}\] Now our system is:\[\begin{align*}9x - 3y &= -12 \x + 3y &= 12\end{align*}\]

Step 3: Add Equations

Now add the two equations to eliminate \(y\):\[\begin{align*}9x - 3y + (x + 3y) &= -12 + 12\end{align*}\]The \(-3y\) and \(3y\) cancel each other out, and we are left with:\[10x = 0\]

Step 4: Solve for x

Divide both sides of the equation \(10x = 0\) by 10 to solve for \(x\):\[x = 0\]

Step 5: Substitute to Solve for y

Substitute \(x = 0\) back into one of the original equations to solve for \(y\). Use the second equation:\[x + 3y = 12\]Substitute \(x = 0\):\[0 + 3y = 12\]Divide both sides by 3:\[y = 4\]

Step 6: Write the Solution

The solution to the system of equations is the point \((x, y)\) which is \((0, 4)\).

Key Concepts

Elimination Method

When solving a system of equations, there are several techniques available, and the elimination method is one of the most structured approaches. This method focuses on eliminating one variable to simplify the problem to a single equation. Here is how it generally works:

• First, you align the system of equations, ensuring that the variables have the same or comparable coefficients.
• Next, you manipulate one or both equations through multiplication or division to make the coefficients of a particular variable identical (but usually opposite in sign) across the equations.
• Finally, you add or subtract the equations to eliminate that variable entirely.

In our exercise, we chose to eliminate the variable \( y \). By strategically multiplying the first equation, the \(-y\) term could be easily canceled when combined with the \(+3y\) from the second equation. Solving systems through elimination is efficient and combines logical reasoning with algebraic manipulation.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. When we solve equations through systems, algebra allows us to represent and manage unknown values efficiently. Here’s why this is helpful in the elimination method:

• It uses variables (like \( x \) and \( y \)) to stand in for values we want to find.
• Equations are solved using operations like addition, subtraction, multiplication, and division to isolate variables.

In the given problem, algebraic operations allowed the equation \( 10x = 0 \) to be solved in a straightforward manner by dividing both sides by 10 to isolate \( x \). By understanding algebraic principles, students can strategically manipulate and resolve complex equations into simple solutions.

Solution of Equations

The final goal in solving a system of equations is to find the values of the variables that satisfy all equations simultaneously. This is called finding the solution of equations:

• A solution must satisfy each individual equation within the system.
• In coordinate geometry, the solution to a two-variable system can often be represented as a point, denoted as \((x, y)\).

In the exercise at hand, the solution to the system was \((0, 4)\), meaning when substituted back into both original equations, these values verify the correctness of our calculations. The process of checking involves substituting back into the initial equations to ensure consistency. Through proper application of elimination and algebraic methods, the solution of equations assures us of mathematical precision.