Question

1.х=12 является корнем уравнения aх-(a∙х/3-3)=1/11 (х-3)+ 5 1/7 х . Найдите a. 2. Решите уравнения: а ) |0,65/(4,75- |0,8 ∙ х-1/3| ) | = 5 ; б) |(|1 59/392|х-9/14-х)|=41/196 . 3. При каких значениях a и b уравнение (a-13) х= b+5 не имеет корней ? ?

Answer 1

1.х=12 является корнем уравнения aх-(a∙х/3-3)=1/11 (х-3)+ 5 1/7 х . Найдите a.

2. Решите уравнения: а ) |0,65/(4,75- |0,8 ∙ х-1/3| ) | = 5 ; б) |(|1 59/392|х-9/14-х)|=41/196 .

3. При каких значениях a и b уравнение (a-13) х= b+5 не имеет корней ?

?

Answer 2

Let's go step by step to solve each of the given equations:

1. To find the value of 'a': Given equation: aх - (a∙х/3 - 3) = 1/11 (х - 3) + 5 1/7 х

Step 1: Simplify the equation: aх - (a∙х/3 - 3) = 1/11 (х - 3) + 36/7 х

Step 2: Get rid of the fractions by multiplying the entire equation by the least common multiple (LCM) of the denominators, which is 77: 77aх - 77(a∙х/3 - 3) = 7(х - 3) + 11 * 36 х

Step 3: Further simplify the equation: 77aх - 77a∙х/3 + 231 = 7x - 21 + 396х

Step 4: Combine 'х' terms on one side and constants on the other side of the equation: 77aх - 77a∙х/3 - 7x + 396х = 231 + 21

Step 5: Factor out 'х' from the left side of the equation: х(77a - 77a/3 + 396 - 7) = 252

Step 6: Simplify the left side: х(154a + 396 - 7) = 252

Step 7: Combine constants: х(154a + 389) = 252

Step 8: Solve for 'х': х = 252 / (154a + 389)

For this equation to have a real solution, the denominator (154a + 389) cannot be zero. Therefore, to avoid division by zero, we have: 154a + 389 ≠ 0

Solve for 'a': 154a ≠ -389 a ≠ -389 / 154

So, 'a' can be any real number except -389 / 154.

2. Solving the equations:

а) |0.65 / (4.75 - |0.8 * x - 1/3|)| = 5

Step 1: Solve the expression inside the absolute value: 4.75 - |0.8 * x - 1/3| = 0.65 / 5

Step 2: Simplify: 4.75 - |0.8 * x - 1/3| = 0.13

Step 3: Move the constant to the other side of the equation:

• |0.8 * x - 1/3| = 0.13 - 4.75

Step 4: Solve for the absolute value expression: |0.8 * x - 1/3| = -4.62

Since the absolute value of any real number is always non-negative, there are no solutions for this equation.

б) |(|1 59/392|x - 9/14 - x)| = 41/196

Step 1: Simplify the expression inside the absolute value: |(|1 59/392|x - 9/14 - x)| = 41/196

Step 2: Combine 'x' terms: |(|1 59/392|x - 9/14)| = 41/196

Step 3: Remove the inner absolute value by considering two cases:

Case 1: (1 59/392)x - 9/14 is positive: (1 59/392)x - 9/14 = 41/196

Step 4: Solve for 'x': (1 59/392)x = 9/14 + 41/196

Step 5: Convert mixed numbers to improper fractions: (557/392)x = (9 * 28 + 41)/196

Step 6: Simplify the right side: (557/392)x = (253/196)

Step 7: Solve for 'x': x = (253/196) * (392/557)

Case 2: (1 59/392)x - 9/14 is negative: (1 59/392)x - 9/14 = -41/196

Step 8: Solve for 'x': (1 59/392)x = 9/14 - 41/196

Step 9: Convert mixed numbers to improper fractions: (557/392)x = (9 * 28 - 41)/196

Step 10: Simplify the right side: (557/392)x = (203/196)

Step 11: Solve for 'x': x = (203/196) * (392/557)

So the two solutions for 'x' are:

x = (253/196) * (392/557) and x = (203/196) * (392/557)

3. For the equation (a - 13)x = b + 5 to have no roots, it means there should be no 'x' value that satisfies the equation. In other words, the equation should represent a horizontal line, which means the coefficient of 'x' (a - 13) should be equal to zero:

a - 13 = 0

Solve for 'a': a = 13

So, when 'a' is equal to 13, the equation (a - 13)x = b + 5 will have no roots. The value of 'b' does not affect the existence of roots in this case. For any value of 'b', the equation will have no solution when 'a' is equal to 13.