solve linear equation w/paper & pencil

Solve:   x - 6 = -12       Solve:   x + 6 = -12       Solve:   x/6 = -12       Solve:   6x = -12 Solve:   - x = -12       Solve:   5x/6 = -12       Solve:   3 + 2x = - 1 Solve:   3 + 2x = 4x - 1      Solve:   3x + 4(x -1) = 2      Solve:   4x + 2 - 3x + 5 - 2(x + 5) = (2x - 5) + (3x + 4) consecutive integer expressions, equations       Solving Equation on Graphing Calculators Intro & Vocabulary   One Operation Equations   More Than One Operation Equations   Problems   More Problems   Solutions to a Linear Equation & Solving Linear Equations Graphically Linear Equation Solver - The Page Does The Work

Intro & Vocabulary

      To solve a linear equation means to find the value(s) of the variable that make(s) the equation true (if this is possible), that make(s) one side or member of the equation equal to the other.

      To solve a linear equation means to find the solution, or answer; to answer, or explain, or to determine the missing information; to cause something not to be a problem anymore.

      To solve a linear equation means to undo, take the inverse of, each operations performed on the variable in order to make/create a statement, equivalent to the original problem, which states that the variable is equal to a specific constant.

Three kinds of solutions are possible when one solves an equation:
• one solution
  - the statement is sometimes true, for example, x = 5 or 2 = y,
• every number is a solution
  - the statement is always true, for example, 5 = 5 or x = x, or,
• there is no solution
  - the statement is false, for example, 10 = 5 or 2 + 3 = 6 + 1.

For a more graphic understanding see Solving Equations Graphically. See the page Bright & Shiny for more examples.

 

 

Undoing One Operation

      The goal of this page is not just to solve the equation -- most of these problems may be solved mentally.   The goal of this page is to write the work and the equivalent statements which solve the equation & justify the answer.

      Here, we solve a linear equation by undoing, taking the inverse of, each operation performed on the variable in order to make/create a statement, equivalent to the original problem, which states that the variable is equal to a specific constant, or a true statement, or a false statement.   Here we show the work.

      In order to do that we answer two questions: "What did they do to the x?" and "How do you undo this?" then, do this to both sides of the equation.

1. Solve:   x - 6 = -12     + 6     + 6     x =   - 6	What did they do to the x? THEY SUBTRACTED 6. How do you undo this? ADD 6.
2. Solve:   x + 6 = -12     - 6     - 6     x =   - 18	What did they do to the x? THEY ADDED 6. How do you undo this? SUBTRACT 6.
3. Solve:       x/6 = -12   (x/6)(6) = (-12)(6)           x = -72	What did they do to the x? THEY DIVIDED BY 6. How do you undo this? MULTIPLY BY 6.
4. Solve:     6x = -12   6x/6 = -12/6       x = -2	What did they do to the x? THEY MULTIPLIED BY 6. How do you undo this? DIVIDE BY 6.

      Two more examples illustrate solving an equation in which one operation must be undone.

      They are isolated here because many students have more difficulty with problems which look like these. Again, the same questions must be addressed: "What did they do to the x?" and "How do you undo this?"

5. Solve:           - x = -12   (- x)/(-1) = (-12)/(-1)               x = 12	What did they do to the x? THEY MULTIPLIED BY -1. How do you undo this? DIVIDE BY -1.
6. Solve:           5x/6 = -12   (5x/6)(6/5) = (-12)(6/5)               x = -72/5	What did they do to the x? THEY MULTIPLIED BY 5/6. How do you undo this? DIVIDE BY 5/6 OR MULTIPLY BY 6/5.

Problems

Solve & show work. Remember:

7.)     9 = x - 12

Answer 9 = x - 12 +12     + 12 _________________ 21 = x

 

8.)     - 3 = - y

Answer     - 3 = - y (- 3)/(-1) = (- y)/(-1)         +3 = y

 

9.)     x + 4 = 20

Answer   x + 4 = 20   -4     -4 ________________   x = 16

 

10.)     2x = 15

Answer   2x = 15 2x/2 = 15/2 x = 15/2

 

 

Solve. MouseOver for answers.

Describe how to undo the operation on x, then state the solution.   ex. 5 above.) Solve -x = -12   11.) Solve:     x - 3 = 1       12.) Solve:     15 = y + 2       13.) Solve:     -4x = -2       14.) Solve:     x/3 = 13       15.) Solve:     2x/3 = 6

More Than One Operation Equations

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