Math  /  Algebra

QuestionLösen Sie die folgenden Gleichungen und führen Sie jeweils die Probe durch. 1.) 3x+7=223 x+7=22 x=5x=5 2.) 7x12=307 \mathrm{x}-12=30 α=6\alpha=6 3.) 8515x=4085-15 x=40 4.) 74x=557-4 x=55

Studdy Solution

STEP 1

1. We are solving linear equations of the form ax+b=cax + b = c.
2. We'll isolate the variable xx in each equation.
3. We verify the solutions by substituting them back into the original equations.

STEP 2

1. Solve the first equation 3x+7=223x + 7 = 22.
2. Verify the solution of the first equation.
3. Solve the second equation 7x12=307x - 12 = 30.
4. Verify the solution of the second equation.
5. Solve the third equation 8515x=4085 - 15x = 40.
6. Verify the solution of the third equation.
7. Solve the fourth equation 74x=557 - 4x = 55.
8. Verify the solution of the fourth equation.

STEP 3

Solve the equation 3x+7=223x + 7 = 22. Subtract 7 from both sides:
3x+77=227 3x + 7 - 7 = 22 - 7 3x=15 3x = 15

STEP 4

Divide both sides by 3 to isolate xx:
x=153 x = \frac{15}{3} x=5 x = 5

STEP 5

Verify the solution by substituting x=5x = 5 back into the original equation:
3(5)+7=15+7=22 3(5) + 7 = 15 + 7 = 22 The left-hand side equals the right-hand side, confirming the solution is correct.

STEP 6

Solve the equation 7x12=307x - 12 = 30. Add 12 to both sides:
7x12+12=30+12 7x - 12 + 12 = 30 + 12 7x=42 7x = 42

STEP 7

Divide both sides by 7 to isolate xx:
x=427 x = \frac{42}{7} x=6 x = 6

STEP 8

Verify the solution by substituting x=6x = 6 back into the original equation:
7(6)12=4212=30 7(6) - 12 = 42 - 12 = 30 The left-hand side equals the right-hand side, confirming the solution is correct.

STEP 9

Solve the equation 8515x=4085 - 15x = 40. Subtract 85 from both sides:
8515x85=4085 85 - 15x - 85 = 40 - 85 15x=45 -15x = -45

STEP 10

Divide both sides by -15 to isolate xx:
x=4515 x = \frac{-45}{-15} x=3 x = 3

STEP 11

Verify the solution by substituting x=3x = 3 back into the original equation:
8515(3)=8545=40 85 - 15(3) = 85 - 45 = 40 The left-hand side equals the right-hand side, confirming the solution is correct.

STEP 12

Solve the equation 74x=557 - 4x = 55. Subtract 7 from both sides:
74x7=557 7 - 4x - 7 = 55 - 7 4x=48 -4x = 48

STEP 13

Divide both sides by -4 to isolate xx:
x=484 x = \frac{48}{-4} x=12 x = -12

STEP 14

Verify the solution by substituting x=12x = -12 back into the original equation:
74(12)=7+48=55 7 - 4(-12) = 7 + 48 = 55 The left-hand side equals the right-hand side, confirming the solution is correct.
Solutions: 1.) x=5x = 5 2.) x=6x = 6 3.) x=3x = 3 4.) x=12x = -12

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