Find three distinct solutions of the equation x1 + 2x2 = 4. Find a parametric representation of the solution set.
Question
Find three distinct solutions of the equation
x_1 + 2x_2 = 4
Find a parametric representation of the solution set.
Solution
Sure, let's solve the equation step by step.
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The given equation is x1 + 2x2 = 4. This is a linear equation with two variables x1 and x2.
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To find distinct solutions, we can choose arbitrary values for one variable and solve for the other.
Solution 1: Let's choose x1 = 0. Substituting x1 = 0 into the equation, we get: 0 + 2x2 = 4 Solving for x2, we get x2 = 4/2 = 2. So, one solution is (x1, x2) = (0, 2).
Solution 2: Let's choose x1 = 2. Substituting x1 = 2 into the equation, we get: 2 + 2x2 = 4 Solving for x2, we get x2 = (4-2)/2 = 1. So, another solution is (x1, x2) = (2, 1).
Solution 3: Let's choose x1 = 4. Substituting x1 = 4 into the equation, we get: 4 + 2x2 = 4 Solving for x2, we get x2 = (4-4)/2 = 0. So, another solution is (x1, x2) = (4, 0).
- Now, let's find a parametric representation of the solution set.
We can rewrite the given equation as x1 = 4 - 2x2.
Let x2 = t (where t is a parameter). Then, x1 = 4 - 2t.
So, the parametric representation of the solution set is (x1, x2) = (4 - 2t, t), where t is any real number.
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