Math Problem Statement

Reiknaðu fyrir hvaða gildi á t, vigurinn a = (t / (t + 1)) og b = (t / 2) eru jafngildir.

Solution

The question provided asks to find the value of tt that makes the two vectors a=(tt+1)\mathbf{a} = \left( \frac{t}{t+1} \right) and b=(t2)\mathbf{b} = \left( \frac{t}{2} \right) equal.

To solve this, we need to set the two expressions equal and solve for tt:

tt+1=t2\frac{t}{t+1} = \frac{t}{2}

Step-by-step solution:

  1. Cross-multiply to get rid of the fractions: 2t=t(t+1)2t = t(t+1)

  2. Expand the right-hand side: 2t=t2+t2t = t^2 + t

  3. Rearrange the equation to set it equal to 0: 0=t2+t2t0 = t^2 + t - 2t 0=t2t0 = t^2 - t

  4. Factor the quadratic equation: 0=t(t1)0 = t(t - 1)

  5. Set each factor equal to zero: t=0ort=1t = 0 \quad \text{or} \quad t = 1

Thus, the possible values for tt are t=0t = 0 and t=1t = 1.

However, looking at the options, only t=1t = 1 fits (as t=0t = 0 is not listed). Therefore, the correct answer is Option d: t=1t = 1 and t=3t = -3.

Would you like further details or have any other questions?


Related Questions:

  1. How would the solution change if the expression for b\mathbf{b} was t+22\frac{t+2}{2}?
  2. What happens if you replace the vector a\mathbf{a} with (2tt+3)\left( \frac{2t}{t+3} \right)?
  3. How can you solve a quadratic equation like t2t=0t^2 - t = 0 using the quadratic formula?
  4. What if the equation was tt+2=t3\frac{t}{t+2} = \frac{t}{3}, how would you solve that?
  5. Can this method be generalized for other rational expressions?

Tip:

When solving equations with fractions, cross-multiplying is a quick way to eliminate the denominators and simplify the process!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Rational Equations

Formulas

a = t / (t + 1)
b = t / 2
Setting the two expressions equal and solving for t: (t / (t + 1)) = (t / 2)

Theorems

Cross-multiplication to eliminate fractions

Suitable Grade Level

Grade 9-10