Before we delve into solving systems of equations, it is essential to review linear equations and variables. A linear equation is an equation that results in a straight line when graphed on a coordinate plane. It can be written in the form of y = mx + b, where m represents the slope of the line, and b represents the y-intercept.

Variables are symbols that represent values that can change. They are used to represent unknowns in algebraic equations. In a linear equation, variables are typically represented by x and y.

It is also important to understand the concept of a solution to a linear equation. A solution is a value that, when substituted for the variable(s) in the equation, makes the equation true. For example, in the equation y = 2x + 1, the solution (2, 5) means that when x = 2 and y = 5, the equation y = 2x + 1 is true.

Now that we have reviewed linear equations and variables, we can move on to solving systems of equations.

In Algebra, we often encounter situations where we need to solve multiple equations at the same time. For example, consider the following scenario:

A school is selling tickets to a play. The tickets cost $5 for students and $8 for adults. The school has sold a total of 300 tickets and has collected $2000. How many tickets of each type have been sold?

To solve this problem, we need to use a system of equations. A system of equations is a set of two or more equations that must be solved simultaneously. In this case, we can define:

x = number of student tickets sold

y = number of adult tickets sold

Then, we can write the following system of equations:

x + y = 300

5x + 8y = 2000

The first equation represents the total number of tickets sold, while the second equation represents the total amount of money collected. To solve this system of equations, we need to find the values of x and y that satisfy both equations.

There are several methods to solve systems of equations, including graphing, substitution, and elimination. In the upcoming sections, we will explore each of these methods in detail and learn how to apply them to solve different types of problems.

One method to solve systems of equations is by using substitution. This involves solving one equation for one variable and substituting the resulting expression into the other equation. The steps to solve a system of equations using substitution are as follows:

1. Select one of the equations and solve for one of the variables in terms of the other variable.
2. Substitute the expression found in step 1 into the other equation and solve for the remaining variable.
3. Substitute the value found in step 2 back into either equation to solve for the other variable.
4. Check your solution by substituting both values into both original equations to ensure they are true.

Let's look at an example:

Solve the system of equations:

2x + 3y = 7
x - y = 2

We can solve the second equation for x in terms of y by adding y to both sides:

x = y + 2

Now, substitute this expression for x in the first equation:

2(y + 2) + 3y = 7

Simplify and solve for y:

2y + 4 + 3y = 7

5y + 4 = 7

5y = 3

y = 3/5

Now, substitute this value of y back into either equation to solve for x:

x - (3/5) = 2

x = 2 + (3/5)

x = 13/5

So the solution to the system of equations is:

x = 13/5, y = 3/5

Check the solution by substituting both values into both original equations:

2(13/5) + 3(3/5) = 7

13/5 - 3/5 = 2

Both equations are true, so the solution is correct.

Another way to solve systems of equations is through elimination. This method involves adding or subtracting the equations to eliminate one of the variables.

Here are the steps to solve a system of equations using elimination:

1. Rearrange the equations so that the variables are in the same order.
2. Multiply one or both equations by a constant so that the coefficients of one variable are opposites.
3. Add or subtract the equations to eliminate one of the variables.
4. Solve for the remaining variable.
5. Substitute the value of the solved variable into either equation to solve for the other variable.

Let's look at an example:

$$\begin{aligned} 2x + 3y &= 7 \\ 4x - 5y &= -6 \end{aligned}$$

In this system, we can eliminate the variable $y$ by multiplying the first equation by $-5$ and adding it to the second equation:

$$\begin{aligned} -10x - 15y &= -35 \\ 4x - 5y &= -6 \end{aligned}$$

Adding the equations, we get:

$$-6x = -41$$

Dividing both sides by $-6$, we get:

$$x = \frac{41}{6}$$

Substituting this value into the first equation, we get:

$$2\left(\frac{41}{6}\right) + 3y = 7$$

Multiplying through by $6$ to eliminate the fraction, we get:

$$41 + 18y = 42$$

Subtracting $41$ from both sides, we get:

$$18y = 1$$

Dividing both sides by $18$, we get:

$$y = \frac{1}{18}$$

Therefore, the solution to the system is:

$$(x,y) = \left(\frac{41}{6}, \frac{1}{18}\right)$$

One of the most important applications of systems of equations is in solving real-life scenarios. For instance, if you are planning to go on a road trip, you might want to know how much gas you need to buy for the journey. In this case, you can use a system of equations to determine the total distance you will travel and the amount of gas you will need.

Let's consider the following example:

You are planning a road trip from New York City to Miami. The distance between these two cities is approximately 1,280 miles. Your car's gas tank can hold up to 15 gallons of gas, and it gets an average of 25 miles per gallon.

To determine how much gas you need to buy for the trip, you can use a system of equations:

d = 1,280

g = d/25

g = 15

Where d represents the total distance of the trip and g represents the amount of gas you need to buy.

By solving this system of equations, you can find that the total amount of gas you need to buy is:

g = d/25 = 1,280/25 = 51.2 gallons

Therefore, you will need to buy approximately 51.2 gallons of gas for your trip.

This is just one example of how systems of equations can be used to solve real-life problems. By understanding the concepts and techniques involved in solving systems of equations, you can apply them to a wide range of scenarios and make better-informed decisions.