Solve the systems and give the coordinates of the solution.
0.2x + 0.2y = 1.2
0.35x - 0.3y = -0.5
(2,4)
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To solve the given system of equations, you can use the method of substitution or elimination. I'll explain both methods briefly, and you can choose the one you prefer.
Method 1: Substitution
1. Rearrange the first equation to solve for one variable in terms of the other.
0.2x + 0.2y = 1.2
0.2x = 1.2 - 0.2y
x = (1.2 - 0.2y) / 0.2
x = 6 - y
2. Substitute this expression for x into the second equation.
0.35x - 0.3y = -0.5
0.35(6 - y) - 0.3y = -0.5
2.1 - 0.35y - 0.3y = -0.5
Combine like terms: -0.65y = -2.6
Divide both sides by -0.65: y = 4
3. Substitute the value of y back into either of the original equations to find the corresponding value of x.
Using the first equation: 0.2x + 0.2(4) = 1.2
0.2x + 0.8 = 1.2
0.2x = 0.4
x = 2
Therefore, the solution to the system of equations is x = 2 and y = 4. The coordinates of the solution are (2, 4).
Method 2: Elimination
1. Multiply the first equation by 5 and the second equation by 10 to make the coefficients of x equal.
5(0.2x + 0.2y) = 5(1.2)
10(0.35x - 0.3y) = 10(-0.5)
Simplify the equations:
x + y = 6
3.5x - 3y = -5
2. Multiply the first equation by -3.5 and add it to the second equation to eliminate x.
-3.5(x + y) + (3.5x - 3y) = -3.5(6) + (-5)
-3.5x - 3.5y + 3.5x - 3y = -21 - 5
-6.5y = -26
Divide both sides by -6.5: y = 4
3. Substitute the value of y back into either of the original equations to find the corresponding value of x.
Using the first equation: x + 4 = 6
x = 2
Therefore, the solution to the system of equations is x = 2 and y = 4. The coordinates of the solution are (2, 4).