In algebra, a variable is a letter or symbol that represents an unknown value or quantity. Variables are essential in algebra because they allow us to solve equations with unknown values. We use variables to represent unknown values in order to find the value of the variable that makes the equation true. For example, let's consider the equation: 2x + 3 = 9 In this equation, x is the variable. We do not know the value of x yet, but we do know that when we substitute the correct value for x, the equation will be true. To solve for x, we need to isolate the variable on one side of the equation. To do this, we can start by subtracting 3 from both sides of the equation: 2x = 6 Now we can divide both sides by 2: x = 3 So, the value of x that makes the equation true is 3. In summary, variables are an important concept in algebra because they allow us to solve equations with unknown values. By representing unknown values with variables, we can manipulate the equation to find the value of the variable that makes the equation true.To solve a linear equation with one variable, we need to isolate the variable on one side of the equation. The following steps will help you solve any linear equation:Step 1: Simplify both sides of the equation by combining like terms.Step 2: Isolate the variable term by adding or subtracting the same value from both sides of the equation.Step 3: Simplify both sides of the equation again by combining like terms.Step 4: Solve for the variable by dividing or multiplying both sides of the equation by the coefficient of the variable.Example:Solve for x: 5x - 7 = 18Step 1: Simplify both sides of the equation by combining like terms.5x - 7 + 7 = 18 + 75x = 25Step 2: Isolate the variable term by adding or subtracting the same value from both sides of the equation.5x/5 = 25/5x = 5Step 3: Simplify both sides of the equation again by combining like terms.x = 5Step 4: Check your answer by substituting the value of x back into the original equation.5(5) - 7 = 1825 - 7 = 1818 = 18Therefore, x = 5 is the correct solution.Remember to always check your answer by substituting it back into the original equation to ensure it is correct.To graph a linear equation, we need to plot at least two points on the coordinate plane and then connect them with a straight line. Here is how to do it step-by-step:1. Write the equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.2. Identify the y-intercept, which is the point where the graph crosses the y-axis. This point has the coordinates (0, b).3. Use the slope to find another point on the line. The slope tells us how much y changes for each unit of x. For example, if the slope is 2/3, then for every increase of 1 in x, y increases by 2/3. To find another point on the line, start at the y-intercept and move vertically up or down by the slope, and horizontally right or left by 1 unit. Repeat this process until you have at least two points.4. Plot the points on the coordinate plane. Label them with their coordinates.5. Draw a straight line through the points. Make sure the line extends beyond the plotted points.6. Check your graph by plugging in other values of x and y to see if they satisfy the equation of the line.Here is an example: Graph the equation y = 2x + 1.1. The equation is already in slope-intercept form, with m = 2 and b = 1.2. The y-intercept is (0, 1).3. To find another point, start at (0, 1) and move up 2 units and right 1 unit to get (1, 3).4. Plot the points (0, 1) and (1, 3).5. Draw a straight line through the points.6. Check the graph by plugging in other values of x and y, such as (2, 5) or (-1, -1).

Remember that the slope of a line can be positive, negative, zero, or undefined. A positive slope means the line goes up as x increases, while a negative slope means the line goes down as x increases. A slope of zero means the line is horizontal, and a undefined slope means the line is vertical.

Graphing linear equations is a fundamental skill in algebra that will be useful in many applications, such as analyzing data or designing structures. Practice graphing different types of lines to become more proficient.

In the previous section, we learned how to solve linear equations with one variable. However, not all problems can be solved with just one equation. Sometimes we need two or more equations to find the values of variables. This is where the concept of systems of equations comes into play.

A system of equations is a collection of two or more equations with the same variables. The solution to a system of equations is the set of values that make all the equations true. To solve a system of equations, we need to find the values of variables that satisfy all the equations simultaneously.

There are different methods to solve a system of equations, but in this section, we will focus on the elimination method. The elimination method involves adding or subtracting the equations in a system to eliminate one of the variables. This results in a new equation with only one variable that can be solved.

Let's take an example to illustrate the elimination method:

Example:

Solve the following system of equations:

2x + 3y = 7

4x - 5y = -11

To use the elimination method, we need to eliminate one of the variables. In this case, we can eliminate y by multiplying the first equation by 5 and the second equation by 3:

10x + 15y = 35

12x - 15y = -33

Now, we can add the two equations:

22x = 2

Dividing both sides by 22, we get:

x = 1/11

Substituting x = 1/11 in any of the original equations, we can find the value of y:

2(1/11) + 3y = 7

3y = 75/11

y = 25/11

Therefore, the solution to the system of equations is (1/11, 25/11).

Practice solving systems of equations using the elimination method with the exercises provided below.

In this section, we will learn how to simplify and solve quadratic equations. A quadratic equation is an equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. To solve a quadratic equation, we need to find the values of x that make the equation true.

The first step in solving a quadratic equation is to simplify it as much as possible. This means combining like terms and putting the equation in standard form, which is ax^2 + bx + c = 0. Once the equation is simplified, we can use several methods to solve it:

Method 1: Factoring

If the quadratic equation can be factored, we can use the zero product property to find the solutions. The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. For example, consider the equation x^2 + 5x + 6 = 0. This equation can be factored as (x + 2)(x + 3) = 0. Using the zero product property, we can see that either (x + 2) = 0 or (x + 3) = 0. Solving for x, we get x = -2 or x = -3. Therefore, the solutions to the equation are x = -2 and x = -3.

Method 2: Completing the Square

If the quadratic equation cannot be factored, we can use the method of completing the square to solve it. Completing the square involves adding and subtracting a constant term to the equation to create a perfect square trinomial. For example, consider the equation x^2 + 6x + 5 = 0. To complete the square, we add and subtract (6/2)^2 = 9 to the equation, giving us:

x^2 + 6x + 9 - 9 + 5 = 0

(x + 3)^2 - 4 = 0

Now we can solve for x by taking the square root of both sides and isolating x:

(x + 3)^2 - 4 = 0

(x + 3)^2 = 4

x + 3 = ±2

x = -3 ± 2

Therefore, the solutions to the equation are x = -1 and x = -5.

Method 3: Quadratic Formula

If neither factoring nor completing the square is possible or desirable, we can use the quadratic formula to solve the equation. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. For example, consider the equation 2x^2 + 3x - 2 = 0. Using the quadratic formula, we get:

x = (-3 ± √(3^2 - 4(2)(-2))) / 2(2)

x = (-3 ± √(25)) / 4

x = (-3 ± 5) / 4

Therefore, the solutions to the equation are x = -1/2 and x = 2.

By using these methods, you can simplify and solve any quadratic equation!