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Solving Linear Equations
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Solving Linear Equations

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Questions and Answers

In the linear equation ax + b = 0, after moving the constant term to the right side, the equation becomes $ax = ______$

-b

After dividing both sides of the equation ax = -b by 'a', the equation becomes $x = ______$

-b / a

In the equation 3x + 5 = 0, after moving the constant term to the right side, the equation becomes $3x = ______$

-5

After dividing both sides of the equation 3x = -5 by 3, the equation becomes $x = ______$

<p>-5 / 3</p> Signup and view all the answers

In the equation ax + bx = cx + d, after combining like terms, the equation becomes $(a + b)x = ______$

<p>c + d</p> Signup and view all the answers

If the given linear equation has multiples of 'x' on both sides, that is, ax + bx = cx + d, you can proceed by replacing ax + bx = cx + d with $(a + b)x = ______$

<p>c + d</p> Signup and view all the answers

Step 2: Move the constant term (______) to the right side of the equation. Replace (a + b)x = ______ with (a + b)x = -\left(______\right).

<p>c + d</p> Signup and view all the answers

Step 3: Divide both sides of the equation by ______. Replace ______x = -\left(c + d\right) with x = -\frac{\left(c + d\right)}{\left(a + b\right)}.

<p>(a + b)</p> Signup and view all the answers

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

<p>(a + b)</p> Signup and view all the answers

Step 2: Move the constant term (______) to the right side of the equation. Replace ax^2 + bx = cx^2 + d with ax^2 + bx = -\left(______\right).

<p>c + d</p> Signup and view all the answers

Step 3: Divide both sides of the equation by ______. Replace ax^2 + bx = -\left(c + d\right) with x^2 = -\frac{\left(c + d\right)}{\left(a + b\right)}.

<p>(a + b)</p> Signup and view all the answers

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

<p>(a + b)</p> Signup and view all the answers

Step 2: Move the constant term (______) to the right side of the equation.

<p>c + d</p> Signup and view all the answers

Step 1: Combine like terms. Replace ______ = cx^3 + d with (a + b)x^2 = c + d.

<p>ax^3 + bx^2</p> Signup and view all the answers

Step 2: Move the constant term (______) to the right side of the equation.

<p>c + d</p> Signup and view all the answers

What is the value of x^2? Substitute the value of ______ and -(c + d) into the expression for x^2.

<p>(a + b)</p> Signup and view all the answers

Study Notes

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.

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Learn about the basic process for solving linear equations and how to handle equations with multiples of x, x^2, or x^3 on both sides. Practice solving linear equations step by step with examples.

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