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## Questions and Answers

What is the first step in applying the substitution method to solve a system of equations?

When applying the substitution method, what is the purpose of substituting x or y back into the second equation?

In graphing linear equations, what do parallel lines on a coordinate plane indicate about a system of equations?

What does the point of intersection of two lines on a coordinate plane represent in a system of linear equations?

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What happens if two lines on a coordinate plane are coincident when representing a system of equations?

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In linear algebra, how is a linear equation in two variables typically written?

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What is the purpose of finding the intersection points of lines represented by linear equations in two variables?

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What does it mean when a system of linear equations with two variables has an infinite number of solutions?

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When does a system of linear equations with two variables become inconsistent?

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Why is it important in real-life scenarios to apply linear equations in two variables through word problems?

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## Study Notes

## Applications of Linear Equations in Two Variables

A linear equation in two variables is a type of equation that represents a relationship between two variables, typically expressed in the form of `ax + by + c = 0`

, where `a`

and `b`

are not equal to 0. Linear equations in two variables are fundamental in various fields of study, including physics, economics, engineering, and computer science. They have numerous practical applications, such as modeling real-life situations, calculating costs, and determining relationships between physical quantities.

## Solving Systems of Linear Equations in Two Variables

In linear algebra, a system of linear equations involves two or more equations with two or more variables each. Given two linear equations in two variables, the system is solvable if the slopes of the corresponding lines are different, meaning nonparallel lines intersect at exactly one point. This intersection point represents the unique solution to the system of equations. When solving a system of linear equations, graphical methods, such as the substitution method, can be applied to determine the solution.

### Substitution Method

The substitution method allows you to solve a system of linear equations by using one equation to solve for one of the variables, then replacing that variable with its value in the other equation. This leads to a single linear equation with one variable, which can be easily solved.

#### Example

For instance, let's consider the system of equations:

```
2x + 5y = 20
3x + 6y = 12
```

To apply the substitution method, we first isolate `x`

in the first equation:

```
2x + 5y = 20
2x = 20 - 5y
x = (20 - 5y) / 2
```

Now, substitute `x`

from the first equation back into the second equation:

```
3x + 6y = 12
3((20 - 5y) / 2) + 6y = 12
```

Simplifying this equation:

```
3(20 - 5y) / 2 + 6y = 12
```

Expanding and simplifying further:

```
3(20 - 5y) / 2 + 6y = 12
3(20 - 5y) = 12 * 2
3(20 - 5y) = 24
```

Now, we can solve for `y`

:

```
3(20 - 5y) = 24
20 - 5y = 8
5y = 12
y = 12 / 5
```

Finally, substitute the value of `y`

back into the equation `x = (20 - 5y) / 2`

to find `x`

:

```
x = (20 - 5(12 / 5)) / 2
x = (20 - 12) / 2
x = 8 / 2
x = 4
```

So, the solution to the given system of linear equations is `(x, y) = (4, 12)`

.

## Graphing Linear Equations

Graphing linear equations provides a visual representation of the relationship between the variables. Each line on a coordinate plane corresponds to a single linear equation. When two lines representing distinct equations intersect, the coordinates of their point of intersection represent the unique solution to the system of equations. If the two lines are parallel or coincident, meaning they have the same slope, the system has no solution.

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## Description

Explore the application of linear equations in two variables, which play a crucial role in various fields like physics, economics, and engineering. Learn how to solve systems of linear equations using methods like the substitution method to find unique solutions. Discover how graphing linear equations visually represents relationships between variables and determines solutions through intersections on a coordinate plane.