Elementary Algebra and Geometry for Schools

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First Degree Equations

A first-degree equation, also known as a linear equation, is an algebraic equation in which the highest exponent of the variable(s) is 1. It can be written in the form:

ax + b = 0

where "x" represents the variable, "a" and "b" are constants, and "a" is not equal to 0. The goal is to solve for the value of "x" that satisfies the equation.

The solutions to a first-degree equation can be found by performing various algebraic operations such as addition, subtraction, multiplication, and division. The solution is typically a single value, but in some cases, there may be no solution or infinitely many solutions.

A first degree equation is one which, when reduced to its simplest form, contains the unknown letter or letters raised to the first power only. Thus the equations 5x -7=18 and 3x + 5x -2 = 34 -x are first degree equations. The equation 2x2 + 7 x -3x -2x2 = 28, as it is written, does not look like a first degree equation, since it contains the unknown raised to the second power. However, when it is written in simplest form by collecting like terms, the two x2 terms disappear, and the equation reduces to 4x = 28. Thus this equation is a first degree equation.

We have already learned to solve first degree equations by adding, subtracting, multiplying, or dividing both members of the equation by the same number. In this pages we will continue to apply these methods to solve equations; however, we will now solve equations which may contain negative as well as positive numbers. Also, we will learn a few "short cuts" that will make our work easier.

Let us state once more the four principles that we have applied in solving equations. These principles apply to equations which contain negative numbers as well as positive numbers. Trese principles are called axioms. An axiom is a statement that is accepted without proof.

 Axioms 1. The same number may be added to both members of an equation ( we call this the" addition axiom"). 2. The same number may be subtracted from both members of an equation (subtraction axiom). 3. Both members of an equation may be multiplied by the same number (multiplication axiom). 4. Both members of an equation may be divided by the same number, providing that the number by which they are divided is not zero (division axiom).

ILLUSTRATIVE EXAMPLES: Applying axioms

1. Solve the equation 6x =-18.

Solution. We are given the value of 6x. To solve the equation, we must find the value of x (that is, the value of 1x); hence we divide both members of the equation by 6: 2. Solve the equation -4x = 22.

Solution. We are given the value of -4x. To solve the equation, we must find the value of x (that is, the value of 1x); hence "ve divide both members of the equation by -4: 3. Solve the equation 6x -7x = 14- 8.

Solution. Collecting like terms, we have -x = 6.

We now have the value of -x (that is, the value of -1x); to find the value of x (that is, the value of + 1x), we must multiply (or divide) both members of the equation by -1: EXERCISES: Equations

Solve each of the following equations: 1.

To solve the equation 5x = -35, we need to isolate the variable x.

Dividing both sides of the equation by 5, we get:

(5x) / 5 = (-35) / 5

x = -7

Therefore, the solution to the equation 5x = -35 is x = -7.

2.

To solve the equation -6x = 54, we can follow the same steps as before to isolate the variable x.

Dividing both sides of the equation by -6, we get:

(-6x) / -6 = 54 / -6

x = -9

Therefore, the solution to the equation -6x = 54 is x = -9.

3.

To solve the equation 7x = -42, we can isolate the variable x by dividing both sides of the equation by 7:

(7x) / 7 = (-42) / 7

x = -6

Therefore, the solution to the equation 7x = -42 is x = -6.

4.

To solve the equation -8x = -32, we can isolate the variable x by dividing both sides of the equation by -8:

(-8x) / -8 = (-32) / -8

x = 4

Therefore, the solution to the equation -8x = -32 is x = 4.

5.

To solve the equation -x = -6, we can isolate the variable x by multiplying both sides of the equation by -1:

(-1)(-x) = (-1)(-6)

x = 6

Therefore, the solution to the equation -x = -6 is x = 6.

6.

To solve the equation 4x = -26, we can isolate the variable x by dividing both sides of the equation by 4:

(4x) / 4 = (-26) / 4

x = -6.5

Therefore, the solution to the equation 4x = -26 is x = -6.5.

7.

To solve the equation 2x = -22, we can isolate the variable x by dividing both sides of the equation by 2:

(2x) / 2 = (-22) / 2

x = -11

Therefore, the solution to the equation 2x = -22 is x = -11.

Answers : 1.-7. 2.-9. 3.-6. 4. 4. 5. 6. 6.-6½. 7. -11. 8. 15. 9. -5. 10. - 4/3 ;. 11.-8. 12. 2. 13. -5/2. 14. -12. 15. -1/5. 16. -3/4. 17. 8. 18.6. 19. -48. 20. -9. 21.4. 22. -5. 23. 3. 24. -10. 25. 8. 26. -11. 27.-3. 28.-13. 29.-18. 30.-14/3. 31. -13. 32. 27. 33. -1/3,. 34. 1. 35. -11.

Transposing terms.

Transposing terms refers to moving terms from one side of an equation to the other side, while changing their sign.

When transposing terms, it's important to maintain the equality of the equation by applying the same operation to both sides. Here are a few examples:

1. Transposing a constant term: Given the equation x + 5 = 10, we can transpose the constant term 5 to the other side by subtracting 5 from both sides:

x + 5 - 5 = 10 - 5

x = 5

2. Transposing a variable term: Given the equation 2x - 3 = 7, we can transpose the variable term 2x to the other side by subtracting 2x from both sides:

2x - 3 - 2x = 7 - 2x

-3 = 7 - 2x

3. Transposing multiple terms: Given the equation 3y + 2 = 8y - 4, we can transpose the terms involving variable y to the other side:

3y - 8y = -4 - 2

-5y = -6

Remember to perform the transposition accurately and maintain the equality throughout the process.

We have already learned how to solve equations of the type: 3x -4 = 8 by adding the same number to both members of the equation. Let us study this method again:

3x -4= 8

Adding 4 to both members 3x -4 +4 = 8 +4

Collecting on the left 3x = 8 +4

Let us pause here to compare what we now have with the original equation. We notice that the only change in the two equations is that the term -4 appeared in the original equation in the left member, and that it has now been changed in the new equation to a positive term, + 4, of the right member. In other words, we can go directly from the form 3x - 4 = 8 to the form 3x = 8 +4 by moving (transposing) the term -4 from the left member to the right member providing we change the sign of the term. When we transpose (move) terms from one member of an equation to the other member of the equation, we are merely applying to the equation the axioms of addition and subtraction which were stated above in this pages.

 A term may be transposed (moved) from one member of an equation to the other member of the equation provided the sign of the term is changed.

ILLUSTRATIVE EXAMPLES: Transposing terms

1. Solve the equation 5x -3 =-18. 2. Solve the equation 27x = 3x -60  EXERCISES: Equations

Solve each of the following equations by transposing the terms containing x to the left member and the terms not containing x to the right member, and by making the correct changes of signs. Answers : 1. 6. 2. -2. 3. -11. 4. -4. 5. 3. 6. 2. 7. -3. 8. -7. 9. 9. 10.-4. 11. 2. 12. 13. 13. -13. 14. -7. 15. 8. 16. 13. 17. -8. 18. 15. 19. -11. 20. 3. 21. 5. 22. 14. 23. -12. 24. -6. 25. 4. 26. 8. 27. -10. 28. -15. 29. 9. 30. 4. 31. 10. 32. 7. 33. 2. 34. -3. 35. 12. 36. 17. 37. 6 38. -3. 39./-15. 40. 9. 41.0. 42. ½. 43. 1/3 . 44. -2/3. 45.1/5. 46.7/8. 47. -3/5. 48. -2/3. 49. 17/2. 50. -17/3. 51.5/4. 52. 4½. 53. - 58/5. 54. -4½. 55. -35/3. 56. 63/8. 57.-1/10. 58. 8/3. 59. -23/6. 60. 7/2.

ILLUSTRATIVE EXAMPLES: Solving equations containing decimals

1. Solve the equation .3x =-1.2.

Solution. We can solve this equation directly by dividing both members by .3, obtaining the solution x =-4. An alternate method of solution is frequently used, however, in which the decimals are first eliminated by multiplying both sides of the equation by 10 (or by a power of 10):

Given equation .3x = -1.2

Multiplying by 10 (multiplication axiom) 3x =-12

Dividing by 3 (division axiom) x = -4

2. Solve the equation 1.5x -16.8 = 2 -.85x.

Solution. To eliminate the decimals, we must multiply .85x by 100; hence we multiply both members by 100. This is the same as multiplying every term by 100:

Given equation 1.5x -1.8 = 2 -.85x

Multiplying by 100 (multiplication axiom) 150x -1680 = 200 -85x

Transposing (addition axiom) 150x + 85x = 200 + 1680

Collecting 235x = 1880

Dividing by 235 (division axiom)

x = 8.

EXERCISES: Solving equations containing decimals

Solve each of the following equations: Answers : 1. 270. 2. 6. 3. 5. 4. -90. 5. 19. 6. 10. 7. 20,000. 8. 6400. 9. 8. 10.-3/4 11. 6. 12. 760.

Summary. The following methods are applied in solving first degree equations:

1. Write the equation.

2. Collect like terms in each member of the equation.

3. Using the rule for transposing terms, rewrite the equation so that every term containing the unknown letter appears on one side of the equality sign and every term that does not contain the unknown letter appears on the other side of the equality sign. (Usually we transpose the terms containing the unknown letter to the left side of the equation, but this is not necessary. We can transpose the terms containing the unknown letter to the right and the other terms to the left.)

4. Collect like terms.

5. Divide both members of the equation by the coefficient of the unknown letter.

ILLUSTRATIVE EXAMPLES: Solving first degree equations

1. Solve the equation 4x -18 + 5x = 22 + 15x -10,

Solution. Given equation 4x -18 + 5x = 22 + 15x -10

Collecting like terms in each member 9x -18 = 15x + 12

Transposing terms containing x to the left and terms not containing x to the right 9x -15x = 12 + 18

Collecting like terms -6x = 30

Dividing both members by -6                x = -5 Answer

Check. Given equation 4x -18 + 5x = 22 + 15x -10

Substituting -5 for x              4(-5) -18 +5(-5) = 22 + 15(-5) -10

Simplifying         -20 -18 -25 = 22 -75 -10

-63 =-63 Check

2. Solve the equation -6x + 17 -3x = 25 -x -8.

Solution. Given equation -6x + 17 -3x = 25 -x -8

Collecting like terms in each member -9x + 17 = 17 -x

Transposing terms containing x to the left and terms not containing x to the right -9x + x = 17-17

Collecting like terms -8x =0

Dividing both members by -8                 x = 0 (Zero divided y -8 is zero.)

Check Given equation       -6x + 17 -3x = 25 -x -8

Substituting 0 for x         -6(0) + 17 -3(0) = 25 -0 -8

Simplifying 0 +17 -0 = 25 -8

17 = 17 Check

EXERCISES: Solving flrst degree equations Answers: 1.-12. 2. -3. 3. 20. 4. -3. 5. -8. 6.  7. 7.  3. 8. -1 . 9. 1. 10. 2. 11. 5. 12. 7. 13. -5. 14. -2. 15. -5. 16. 64/7. 17. -2. 18. 1. 19. 9/2 20. 45/7. 21. 20/3 . 22. 42. 23. 1. 24. 11/5. 25. 3/2 . 26. 4. 27. 9000. 28. 3.

Equations containing parentheses. We have learned to remove parentheses in algebraic expressions such as 2(x +4), -3(x -5), etc. by multiplying each term within the parentheses by the monomial outside; that is

2(x +4) is equal to 2x +8

and              -3(x -5) is equal to -3x + 15

In solving equations containing parentheses, we first remove the parenheses and then we solve the equation by the methods we have been using.

ILLUSTRATIVE EXAMPLE: Solving an equation containing parentheses

Solve the equation 2(2x + 3) -6(x -2) = 24.

Solution. Given equation 2(2x +3) -6(x -2) = 24

Removing parentheses   4x + 6 -6x + 12 = 24

Collecting like terms in the left member -2x +18 = 24

Transposing the term 18 to the right member -2x = 24 -18

Collecting terms in the right member  -2x = 6

Dividing both members by -2 x = -3 Answer

Check. Given equation 2(2x +3) -6(x -2) = 24

Substituting -3 for x         2(-6 +3) -6(-3 -2) = 24

Simplifying        2(-3) -6(-5) = 24

-6 + 30 = 24

24 = 24 Check

EXERCISES: Solving equations containing parentheses Answers : 1.-1. 2.-3. 3. -7. 4. 3. 5.0. 6. - 4/3. 7. -8/5 . 8. -3/2. 9. 3. 10. 11. 11. 1. 12. 2. 13.0. 14. 13. 15. 2. 16. 10. 17.0. 18. 6. 19. -1. 20. 1/10

Solve x:     (2 + x) -(3 -x) = 7

(2 + x) -(3 -x) = 7

2 +x -3 +x = 7

2x -1 =7

2x = 6