Problem 22 - Algebra 1

Question

Choose a method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{aligned} &3 x+6 y=8\\\ &-6 x+3 y=2 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2/3\) and \(y = 10/9\).

Step 1: Arrange the equations for elimination

Align the like terms vertically. The system is already arranged in this form: \(3x + 6y = 8\) and \(-6x + 3y = 2\).

Step 2: Add the two equations

To eliminate x, add \((3x + 6y = 8)\) and \((-6x + 3y = 2)\) together. This gives us \(9y = 10\).

Step 3: Solve for y

To find the value of y, divide both sides of the equation by 9. This yields \(y = 10/9\).

Step 4: Substitute y into one of the original equations

Substitute y = 10/9 into the first equation \(3x + 6y = 8\). This yields the equation \(3x + 60/9 = 8\).

Step 5: Solve for x

To find the value of x, subtract 60/9 from both sides and divide the result by 3, this gives \(x = 72/90 - 60/90 = 12/90 = 2/3\).

Key Concepts

Understanding Linear Equations

Linear equations are an essential cornerstone in algebra and mathematics. A linear equation looks like this: \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In linear equations, the power of the variable is always one. This means you don't see squares or cubes, just plain variables.
Linear equations can be graphed as straight lines on a coordinate plane. They represent consistent rates of increase or decrease. Furthermore, when you have two linear equations, you form a linear system. Solving this system is about finding a point where these two lines intersect.
There are a few methods to solve linear systems, each with unique steps and ideas. It's all about choosing the best method for your specific problem. Whether it's elimination or substitution, both offer pathways to find the intersection point of two lines.

Exploring the Elimination Method

Elimination is a strategic method in algebra for solving systems of linear equations. The goal with elimination is to remove one of the variables by combining the equations.
To start, you align your equations vertically, ensuring similar terms like x's and y's line up straight. Often, you might need to multiply one or both of the equations by a constant to set up a situation where adding or subtracting the equations will cancel out one of the variables.
In this exercise, we added the two equations:

• From: \(3x + 6y = 8\) and \ \(-6x + 3y = 2\)
• To: \(9y = 10\)

This step elegantly cancels out x, making it easier to solve for y first. It's a method of simplification that requires practice to master but is incredibly powerful for dealing with linear systems.

Embracing the Substitution Method

The substitution method is another handy tool for solving linear systems. It involves expressing one variable in terms of the other in one of the equations, then substituting this back into the second equation.
You start by solving one of the equations for one variable, say x, in terms of y. Then, take this expression and substitute it wherever x appears in the other equation. This will leave you with only one variable in the second equation, which you can solve like a single linear equation.
Although this wasn’t used as the primary method in our example, substitution can be especially useful when one equation is already solved for one variable, making the process quick and straightforward. It's often taught alongside elimination because each method has its unique advantages depending on the problem at hand.