Solve the following system of equations and find the union of the solutions: \(2x + 3y = 6\) and \(5x - y = 10\)
Answer
The union of the solutions is the set of all solutions. Since there is only one solution to this system of equations, the union is simply the set \(\{\frac{36}{7}, -\frac{6}{7}\}\)
Step 1 ：First, isolate y in both equations. For the first equation, we get \(y = \frac{6-2x}{3} = 2-\frac{2x}{3}\). For the second equation, we get \(y = 5x - 10\)
Step 2 ：Then, set these two expressions for y equal to each other and solve for x: \(2-\frac{2x}{3} = 5x - 10\). This simplifies to \(\frac{5x + 2x}{3} = 12\), then \(x = \frac{12*3}{7} = \frac{36}{7}\)
Step 3 ：Substitute \(x = \frac{36}{7}\) into the first equation, we get \(2*\frac{36}{7} + 3y = 6\), then solve for y, we get \(y = \frac{6 - 2*\frac{36}{7}}{3} = \frac{6*7-2*36}{21} = -\frac{18}{21} = -\frac{6}{7}\)
Step 4 ：So the solutions to the system of equations are \(x = \frac{36}{7}\), \(y = -\frac{6}{7}\)
Step 5 ：The union of the solutions is the set of all solutions. Since there is only one solution to this system of equations, the union is simply the set \(\{\frac{36}{7}, -\frac{6}{7}\}\)
