System of Equations

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What is a System of Equations?

A system of equations is simply a set of two or more equations with two or more unknowns. We need to find the value of the unknowns that make all the equations in the system true.

Example: Consider the following system:

1. x+y=5
2. x−y=1

Here, we have two equations and two unknowns, $x$ and $y$.

How to Solve a System of Equations?

There are several methods to solve them:

• Graphical Method: Plotting both equations on a graph and finding the point of intersection.
• Substitution Method: Solve one equation for one variable in terms of the other variable and substitute that expression into the other equation.
• Elimination Method (or Addition Method): Add or subtract the equations to eliminate one variable, making it easier to solve.

Further there is a focus on the substitution and elimination methods.

Substitution Method

Let’s solve our earlier example using the substitution method:

1. From the first equation $x+y=5$, we can express $y$ in terms of $x$ as: $y=5-x$
2. Now, we’ll substitute this expression for $y$ into the second equation $x-y=1$: $x-(5-x)=1$ Solving this, we get: $x=3$
3. Putting $x=3$ in our expression for $y$, we get: $y=2$

So, $x=3$ and $y=2$ is the solution for the system.

Elimination Method

Let’s solve the same system using the elimination method:

• To eliminate $y$, we’ll add both equations: $x+y+x-y=5+1$
  • 2$x=6$
  • $x=3$
• To find $y$, we can substitute $x=3$ in one of the original equations: $3+y=5$
  • $y=2$

Again, we get $x=3$ and $y=2$.

Related Questions:

• Q: Why do we have to solve for two variables? A: In a system with two equations, we generally have two unknowns. To pinpoint a unique solution, we need to find the values of both these unknowns. If you have more equations, you might have more variables to solve for!

 

• Q: Can all systems be solved? A: Great question! Not all systems have a unique solution. Some might have no solutions, while others might have infinitely many. For instance, two overlapping lines have infinite solutions, and two parallel lines have no common solution.

 

• Q: How do I know which method to use? A: With practice, you’ll find one method might be more straightforward for certain problems. However, both methods are valid and can solve any system of linear equations.

Problems to solve:

1. Solve the following system using any method: a) $2x+y=8$ b) $x-2y=3$
2. Solve this system: a) $3x+4y=12$ b) $x+y=3$

Solutions:

• Using elimination, we can multiply the second equation by 2 to get: a) $2x+y=8$ b) $2x-4y=6.$Subtracting the second from the first, we get: $y=2$ $y=2/5.$Substituting this value into the first equation, we get: $x=15/4$

 

• Using substitution, from the second equation $x=3-y.$Substituting in the first: $3(3-y)+4y=12.$ Solving, we get: $y=1.5$ and $x=1.5$

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