Solving Simultaneous Linear Equations in Two Variables

Solving Simultaneous Linear Equations in Two Variables

When you're dealing with more than one line on a graph, things can get interesting! In this world of simultaneous linear equations, we encounter pairs of lines where solving each equation individually wouldn't reveal their unique point(s) of intersection — the solution we seek. Let's dive into understanding how we tackle these simultaneous equations using two primary methods: Elimination Method and Substitution Method, followed by some practical examples from everyday life.

Elimination Method

The Elimination Method focuses on getting rid of one variable within both equations so that only one variable remains in either equation, thus making it easier to solve for the other variable. For example, let's consider (x + y = 7) and (3x - y = 9).

To eliminate (y), multiply the first equation by (-3):

[-3(x+y)=-21] [ -3x-3y=-21 ]

Now, add the modified form of the first equation ((-3x-3y)) to the second equation ((3x-y=9)):

[(3x-y)+ (-3x-3y) = 9 + (-21)] [\underline{-4}y=-12]

Divide all terms by (-4) to isolate the remaining variable:

[y = \frac{1}{-4}(-12)] [y=3]

Finally, substitute the value of (y) back into any original equation to find (x):

[x+3=7] [ x=4]

So, our solution is ((x, y) = (4, 3)). This process works because when choosing which variable to eliminate, we maintain consistency between our multiplication and addition operations; otherwise, we could end up creating new variables instead.

Substitution Method

In contrast to the Elimination Method, the Substitution Method involves directly finding a value for one variable, plugging it into either equation, and then solving for the other variable. To illustrate, consider the following set of equations:

[3x+2y=18] [2x+y=8]

First, solve for (y) in one of the equations:

From the first equation, subtract (3x) from both sides: [2y=18-3x]

Next, divide through by (2): [y=\frac{1}{2}(18-3x)]

Substitute this expression for (y) into another equation:

[2x+\left(\frac{1}{2}(18-3x)\right)=8]

Simplify the left side: [2x+\frac{1}{2}\cdot 18-\frac{3}{2}x=8]

Combine like terms: [x+\frac{9}{2}-\frac{3}{2}x=8]

Solve for (x): [\frac{1}{2}x=\frac{17}{2}] [x=17]

Finally, substitute the found value of (x) back into the expression for (y):

[y=\frac{1}{2}(18-(17))] [y=1]

Our solution is ((x, y) = (17, 1)). Note that this method doesn't guarantee uniqueness since sometimes there exist multiple solutions.

Real Life Applications

Both the Elimination and Substitution Methods have numerous real-life applications involving systems of linear equations such as:

• Finding the distance and direction from one place to another on a map.
• Determining profit margins based on production costs and sales prices.
• Calculating interest over time given initial investment amounts and annual rates.
• Analyzing data trends related to population growth, market demand, and resource distribution.

By applying these techniques, we unlock valuable insights into the relationships among different factors in various fields, ultimately leading to better decision-making processes.