What is the solution for the variables q and r in the system of linear equations: 12q + 3r = 15 and 4q + 4r = 44?

Solution to the System of Linear Equations

To find the values of variables q and r from the following system of linear equations:

• Equation 1: 12q + 3r = 15
• Equation 2: 4q + 4r = 44

let’s solve them step by step.

Step 1: Simplify Equations

First, we can simplify Equation 2 for better manipulation:

• Divide every term by 4:
• q + r = 11 (This is now our Equation 3)

Step 2: Express one variable in terms of the other

From Equation 3, we can express r in terms of q:

• r = 11 – q

Step 3: Substitute into Equation 1

Now, substitute the expression for r back into Equation 1:

• 12q + 3(11 – q) = 15

Step 4: Solve for q

Distributing the 3 gives:

• 12q + 33 – 3q = 15

Simplifying further:

• 9q + 33 = 15
• 9q = 15 – 33
• 9q = -18
• q = -2

Step 5: Substitute q back to find r

Now that we have q = -2, we can substitute this value back into our Equation 3:

• r = 11 – (-2)
• r = 11 + 2
• r = 13

Final Answer

Therefore, the solution to the system of equations is:

• q = -2
• r = 13

We can verify by plugging these values back into the original equations to ensure they hold true.

Verification

Plugging into Equation 1:

• 12(-2) + 3(13) = -24 + 39 = 15, extbf{True!}

Plugging into Equation 2:

• 4(-2) + 4(13) = -8 + 52 = 44, extbf{True!}

Both equations are satisfied, confirming our solution is correct.