4x + 3y = 6 -4x + 2y = 14 Solve the system of equations.

The Correct Answer and Explanation is:

We have the following system of equations:

4x+3y=64x + 3y = 64x+3y=6
−4x+2y=14-4x + 2y = 14−4x+2y=14

We will solve this system using the elimination method. The idea is to manipulate the equations such that one of the variables cancels out when the equations are added or subtracted.

Step 1: Eliminate one variable

Looking at the two equations, notice that the coefficients of xxx in both equations are opposites (4 and -4). We can eliminate xxx by adding both equations.

Adding the two equations:

(4x+3y)+(−4x+2y)=6+14(4x + 3y) + (-4x + 2y) = 6 + 14(4x+3y)+(−4x+2y)=6+14

This simplifies to:(4x−4x)+(3y+2y)=20(4x – 4x) + (3y + 2y) = 20(4x−4x)+(3y+2y)=200x+5y=200x + 5y = 200x+5y=20

Now we have:5y=205y = 205y=20

Step 2: Solve for yyy

To solve for yyy, divide both sides of the equation by 5:y=205=4y = \frac{20}{5} = 4y=520=4

Step 3: Substitute y=4y = 4y=4 into one of the original equations

Now that we know y=4y = 4y=4, we can substitute this value back into one of the original equations to solve for xxx. Let’s use the first equation:4x+3y=64x + 3y = 64x+3y=6

Substitute y=4y = 4y=4 into this equation:4x+3(4)=64x + 3(4) = 64x+3(4)=64x+12=64x + 12 = 64x+12=6

Now, subtract 12 from both sides:4x=6−124x = 6 – 124x=6−124x=−64x = -64x=−6

Divide by 4:x=−64=−32x = \frac{-6}{4} = -\frac{3}{2}x=4−6=−23

Final Solution:

The solution to the system of equations is:x=−32,y=4x = -\frac{3}{2}, \quad y = 4x=−23,y=4

Explanation:

We used the elimination method to eliminate xxx and solve for yyy. Then, we substituted y=4y = 4y=4 into one of the original equations to find x=−32x = -\frac{3}{2}x=−23. The solution is (−32,4)\left( -\frac{3}{2}, 4 \right)(−23,4).