Final answer:
To solve the given system of equations, we can use the method of elimination. By multiplying the equations and eliminating x and y, we find that x = 3.857 and y = 0.715.
Step-by-step explanation:
To solve the system of equations 0.3x + 0.2y = 1.3 and 0.5x + 0.4y = 2.2, we can use the method of elimination. Multiply the first equation by 2 and the second equation by 3 to create equal coefficients for y in both equations. This gives us 0.6x + 0.4y = 2.6 and 1.5x + 1.2y = 6.6. Subtract the first equation from the second equation to eliminate x: (1.5x + 1.2y) - (0.6x + 0.4y) = 6.6 - 2.6. This simplifies to 0.9x + 0.8y = 4.
Next, solve the resulting equation for x or y. Let's solve for x by multiplying the first equation by 0.9 to make the coefficients of x the same: 0.9(0.3x + 0.2y) = 0.9(1.3). This gives us 0.27x + 0.18y = 1.17. Subtract this new equation from the equation 0.9x + 0.8y = 4: (0.9x + 0.8y) - (0.27x + 0.18y) = 4 - 1.17. This simplifies to 0.63x + 0.62y = 2.83.
Now, we have a new system of equations: 0.9x + 0.8y = 4 and 0.63x + 0.62y = 2.83. Use the elimination method again to eliminate y. Multiply the first equation by 0.62 and the second equation by 0.8 to make the coefficients of y the same. This gives us 0.56x + 0.496y = 2.48 and 0.504x + 0.496y = 2.264.
Subtract the first equation from the second equation to eliminate x: (0.504x + 0.496y) - (0.56x + 0.496y) = 2.264 - 2.48. This simplifies to -0.056x = -0.216. Solve for x by dividing both sides of the equation by -0.056: x = 3.857.
Now substitute this value of x into one of the original equations to find y. Using the equation 0.3x + 0.2y = 1.3, we have 0.3(3.857) + 0.2y = 1.3. Simplify the equation: 1.157 + 0.2y = 1.3. Subtract 1.157 from both sides: 0.2y = 0.143. Solve for y by dividing both sides by 0.2: y = 0.715.
Therefore, the solution to the system of equations is x = 3.857 and y = 0.715.