Solving Linear Equations

Solving Linear Equations

Linear equations play a fundamental role in mathematics as they are used to represent relationships between two variables. These equations can be written in the form ax + b = 0, where a and b are constants, and x is the variable. In this section, we will discuss how to solve linear equations for x.

The Basic Process for Solving Linear Equations

1. Step 1: Move the constant term (b) to the right side of the equation.

   Replace the original equation ax + b = 0 with ax = -b.

2. Step 2: Divide both sides of the equation by a.

   Replace ax = -b with x = -b / a.

3. Step 3: Answer the question "What is the value of x?"

   Substitute the value of a and b from the original equation into the expression for x.

For example, consider the equation 3x + 5 = 0.

1. Move the constant term (5) to the right side of the equation: 3x = -5.
2. Divide both sides by 3: x = -5 / 3.
3. Substitute the values of a and b into the expression for x: x = -5 / 3.

The value of x is -5 / 3.

Solving Linear Equations with Multiples of x on Both Sides

If the given linear equation has multiples of x on both sides, that is, ax + bx = cx + d, you can proceed as follows:

1. Step 1: Combine like terms.

   Replace ax + bx = cx + d with (a + b)x = c + d.

2. Step 2: Move the constant term (c + d) to the right side of the equation.

   Replace (a + b)x = c + d with (a + b)x = -(c + d).

3. Step 3: Divide both sides of the equation by (a + b).

   Replace (a + b)x = -(c + d) with x = -(c + d) / (a + b).

4. Step 4: Answer the question "What is the value of x?"

   Substitute the value of (a + b) and -(c + d) into the expression for x.

For example, consider the equation 2x^2 + 3x = 5x - 4.

1. Combine like terms: 2x^2 + 3x = 5x - 4 becomes x^2 + 3x = 5x - 4.
2. Move the constant term (5x - 4) to the right side of the equation: x^2 + 3x = -(5x - 4).
3. Divide both sides by (1 + 3): x = -(5x - 4) / (1 + 3) = -(5x - 4) / 4.
4. Substitute the values of (1 + 3) and -(5x - 4) into the expression for x: x = -(5x - 4) / 4.

The value of x is -(5x - 4) / 4.

Solving Linear Equations with Multiples of x^2 on Both Sides

If the given linear equation has multiples of x^2 on both sides, that is, ax^2 + bx = cx^2 + d, you can proceed as follows:

1. Step 1: Combine like terms.

   Replace ax^2 + bx = cx^2 + d with (a + b)x^2 = c + d.

2. Step 2: Move the constant term (c + d) to the right side of the equation.

   Replace (a + b)x^2 = c + d with (a + b)x^2 = -(c + d).

3. Step 3: Divide both sides of the equation by (a + b).

   Replace (a + b)x^2 = -(c + d) with x^2 = -(c + d) / (a + b).

4. Step 4: Answer the question "What is the value of x^2?"

   Substitute the value of (a + b) and -(c + d) into the expression for x^2.

For example, consider the equation 2x^2 + 3x = 5x^2 - 4.

1. Combine like terms: 2x^2 + 3x = 5x^2 - 4 becomes 3x^2 = 2x - 4.
2. Move the constant term (2x - 4) to the right side of the equation: 3x^2 = -(2x - 4).
3. Divide both sides by 3: x^2 = -(2x - 4) / 3.
4. Substitute the values of 3 and -(2x - 4) into the expression for x^2: x^2 = -(2x - 4) / 3.

The value of x^2 is -(2x - 4) / 3.

Solving Linear Equations with Multiples of x^3 on Both Sides

If the given linear equation has multiples of x^3 on both sides, that is, ax^3 + bx^2 = cx^3 + d, you can proceed as follows:

1. Step 1: Combine like terms.

   Replace ax^3 + bx^2 = cx^3 + d with (a + b)x^2 = c + d.

2. Step 2: Move the constant term (c + d) to the right side of the equation.