Question

question
solve the system of equations using elimination: $4x - 2y = 10$ and $7x - 4y = 13$.

Explanation:

Step1: Multiply first equation by 2

To eliminate \( y \), we multiply the first equation \( 4x - 2y = 10 \) by 2. This gives us \( 2(4x - 2y)=2\times10 \), which simplifies to \( 8x - 4y = 20 \).

Step2: Subtract the second equation

Now we have two equations: \( 8x - 4y = 20 \) (from Step 1) and \( 7x - 4y = 13 \) (original second equation). Subtract the second equation from the first: \( (8x - 4y)-(7x - 4y)=20 - 13 \). Simplifying the left side: \( 8x - 4y - 7x + 4y = x \), and the right side is \( 7 \). So \( x = 7 \).

Step3: Substitute x into first equation

Substitute \( x = 7 \) into the first original equation \( 4x - 2y = 10 \). We get \( 4\times7 - 2y = 10 \), which is \( 28 - 2y = 10 \).

Step4: Solve for y

Subtract 28 from both sides: \( -2y = 10 - 28 = -18 \). Then divide both sides by -2: \( y=\frac{-18}{-2}=9 \).

Answer:

The solution to the system is \( x = 7 \) and \( y = 9 \), or as an ordered pair \( (7, 9) \).