Linear equations are mathematical expressions that show a linear relationship between variables. They form a straight line on a graph and can have one, two, or three variables. Solving linear equations involves finding the values of the variables that satisfy the equation.

There are three commonly used forms of linear equations:

**Standard Form:**In this form, the equation is written as a sum of x and y with a constant. Examples include:

- One-variable linear equation:
- Two-variable linear equation:
- Three-variable linear equation:

**Slope-Intercept Form:**This form is written as y = mx + b, where m is the slope and b is the y-intercept. Example:

**Point Slope Form:**This form uses coordinates on the plane to create a straight line equation.

**Function Form:**This form replaces y with f(x), with x as the independent variable and f(x) as the dependent variable.

Given two points, we can write a linear equation for the line passing through them. The first step is finding the slope or gradient, given by the formula:

Using coordinates (2, 8) and (4, 3) as an example, we get a slope of . To find the y-intercept, we use the standard form equation:

From this equation, we determine the y-intercept to be 7, and the equation for the line is:

Linear equations can also be used to solve word problems. When faced with such problems, follow these steps:

- Familiarize with the problem and its context.
- Identify variables and what they represent in the problem.
- Write an equation that represents the given information.
- Solve the equation to find the values of the variables.

For example, consider this word problem:

To solve this, we can follow the steps outlined above:

- Familiarize with the problem: Tickets to a music show cost $162 for 12 kids and 3 adults. On the same show, 8 kids and 3 adults also spent $122 on tickets.
- Identify variables: Let x represent the cost for kids and y represent the cost for adults.
- Write an equation: We can create two equations based on the given information:
- Solve the system of equations to find the values of x and y.

When faced with a problem, it is crucial to break it down into smaller parts. For example, consider the scenario where 12 kids and 3 adults spent $162, while 8 kids and 3 adults spent $122. To solve this, we must identify the variables in the equation. Let x represent the cost of kids' tickets and y represent the cost of adults' tickets.

The equation for 12 kids + 3 adults is $162 and the equation for 8 kids + 3 adults is $122. These types of equations are known as simultaneous equations. We can use either substitution or the elimination method to find the values of x and y, with the elimination method being used in this case.

To start, we subtract the second equation from the first, giving us:

4x=$40

By dividing both sides by 4, we can solve for x, giving us x=$10. To find the value of y, we can substitute x=$10 into any of the original equations. For this example, we will use the equation for 8 kids + 3 adults, giving us:

$80 + 3y=$122

By subtracting $80 from both sides and dividing by 3, we find that y=$14. Therefore, we now know that a ticket costs $10 for kids and $14 for adults. Remember, we let x represent kids' tickets and y represent adult tickets.

To effectively write linear equations, it's crucial to grasp their properties. Linear equations are algebraic functions that can be depicted as straight lines on a Cartesian plane, with points on the line representing x and y values. To determine the slope of a line parallel to another, we must first put the equation in standard form.

This requires making y the subject of the equation, giving us:

y=2x+1

The slope, or coefficient of x, can now be easily identified as 2. Therefore, the equation for the line parallel to this one is:

y=2x+b

To find the value of b, we can substitute the given point (3,0) into the equation and solve for b.

0=2(3)+b

By simplifying, we get b=-6. Hence, the equation for the parallel line is:

y=2x-6

- Linear equations can be represented as straight lines on a Cartesian plane.
- When dealing with two points, the slope can be calculated using the formula m=(y2-y1)/(x2-x1).
- The standard form of a linear equation is Ax+By=C, where A and B are coefficients and x and y are variables.

So, how exactly do we write a linear equation? The most common method is by using the slope-intercept form:

y=mx+b

Here, y is the y-coordinate on the graph, m is the slope, x is the x-coordinate, and b is the y-intercept.

When provided with two points, we can determine the slope using the formula m=(y2-y1)/(x2-x1). Then, we plug in the slope and one of the points into the standard form of a linear equation and solve for the y-intercept.

The standard form for a linear equation is Ax+By=C, where A and B are coefficients and x and y are variables. To write an equation in this form, we must manipulate the given equation by rearranging terms and simplifying.

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