Solving Pairs of Linear Equations in Two Variables

Solving Pairs of Linear Equations in Two Variables

When you encounter two linear expressions involving the same pair of variables (let's say (x) and (y)), their intersection usually represents a point where both equations hold true simultaneously. This section will delve into how we tackle pairs of such linear equations. We'll specifically explore graphical methods using coordinate planes, along with algebraic techniques like substitution and elimination.

Graphing Method

To visualize solutions, you can create a coordinate plane, plotting each equation by connecting points represented by its ordered pairs ((x, y)):

1. Plot each line individually based on (y = mx + b) (the slope-intercept form), where (m) is the slope, and (b) is the (y-) intercept.
2. Identify the points of intersection between lines when they cross. These locations represent all possible values of (x) and (y) satisfying both equations simultaneously.

Substitution Method

Given one of the equations, solve it for either variable — let's assume (x). Then substitute this expression into another equation, forming a single equation in only one remaining variable, typically (y). Solve that simplified equation, which yields exact or approximate values for (y). Finally, plug these back into the original (x=)formula from step 1 to find corresponding values for (x).

For example, consider the following system: [ \begin{aligned} & x - y = 1 \ & 2x + y = 7 \end{aligned} ]

First, rewrite the first equation as (x=y+1.) Now substitute this into the second equation: (2(y + 1) + y = 7,) which becomes (3y = 4;) therefore, (y = \frac{4}{3}.) Next, plug this value of (y) back into our initial replacement formula ((x=y+1),) yielding (x =\frac{4}{3}+1 = \frac{7}{3}). Thus, the solution is (\left(\dfrac{7}{3}, \dfrac{4}{3}\right)).

Elimination Method

Another approach involves manipulating the coefficients so that one variable disappears entirely, leaving behind only one equation in terms of the other. For instance, if you multiply the first equation by a certain constant and then add it to the second equation, the coefficient of some variable may cancel out—leaving us with a single equation in only one unknown.

Suppose we have the same system of equations mentioned above: [ \begin{aligned} & x - y = 1 \ & 2x + y = 7 \end{aligned} ]

Multiplying the first equation by 2 gives us (2x - 2y = 2.) Adding this to the second equation results in (3x = 9;$ thus, $x = 3.$ Subsequently, finding (y) via the first equation (since (x) has already been solved): (y = x - 1 = 2.) So, once again, the unique solution is (\left(3, 2\right)).

In conclusion, there exist multiple approaches to solving systems of linear equations, including graphical analysis alongside more traditional algebraic strategies. Each technique offers unique insights and practical applications depending on the specific problem at hand.