Math  /  Algebra

Question0.8x2.1y=10.81.6x0.7y=7.6\begin{array}{l}0.8 x-2.1 y=10.8 \\ 1.6 x-0.7 y=7.6\end{array}

Studdy Solution

STEP 1

1. The problem involves solving a system of linear equations with two variables, xx and yy.
2. We will use the method of elimination to solve the system of equations.
3. The coefficients and constants are real numbers.

STEP 2

1. Multiply the equations by appropriate factors to align coefficients for elimination.
2. Subtract one equation from the other to eliminate one variable.
3. Solve for the remaining variable.
4. Substitute the solved value back into one of the original equations to find the other variable.
5. Verify the solution by substituting both values into the original equations.

STEP 3

Multiply the first equation by 1.61.6 to align the coefficients of xx: 1.6×(0.8x2.1y)=1.6×10.8 1.6 \times (0.8x - 2.1y) = 1.6 \times 10.8 1.28x3.36y=17.28 1.28x - 3.36y = 17.28

STEP 4

Multiply the second equation by 0.80.8 to align the coefficients of xx: 0.8×(1.6x0.7y)=0.8×7.6 0.8 \times (1.6x - 0.7y) = 0.8 \times 7.6 1.28x0.56y=6.08 1.28x - 0.56y = 6.08

STEP 5

Subtract the second modified equation from the first modified equation to eliminate xx: (1.28x3.36y)(1.28x0.56y)=17.286.08 (1.28x - 3.36y) - (1.28x - 0.56y) = 17.28 - 6.08 3.36y+0.56y=11.20 -3.36y + 0.56y = 11.20 2.80y=11.20 -2.80y = 11.20

STEP 6

Solve for yy: y=11.202.80 y = \frac{11.20}{-2.80} y=4 y = -4

STEP 7

Substitute y=4y = -4 back into the first original equation to solve for xx: 0.8x2.1(4)=10.8 0.8x - 2.1(-4) = 10.8 0.8x+8.4=10.8 0.8x + 8.4 = 10.8 0.8x=2.4 0.8x = 2.4 x=2.40.8 x = \frac{2.4}{0.8} x=3 x = 3

STEP 8

Verify the solution (x,y)=(3,4)(x, y) = (3, -4) by substituting into the second original equation: 1.6(3)0.7(4)=7.6 1.6(3) - 0.7(-4) = 7.6 4.8+2.8=7.6 4.8 + 2.8 = 7.6 7.6=7.6 7.6 = 7.6 Since both original equations are satisfied, the solution is confirmed.
The solution to the system of equations is x=3x = 3 and y=4y = -4.

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