Solve the system of equations:
$\begin{array}{l}4 x+3 y=29 \\ 6 x+7 y=41\end{array}$
We are given a system of two linear equations with two variables, \(x\) and \(y\):
\[ \begin{array}{l} 4x + 3y = 29 \quad \text{(Equation 1)} \ 6x + 7y = 41 \quad \text{(Equation 2)} \end{array} \]
Our objective is to determine the unique values of \(x\) and \(y\) that satisfy both equations. A system of linear equations can be solved using several methods, such as substitution, elimination, or matrix operations. In this case, the elimination method is particularly efficient. We will manipulate the equations to eliminate one variable by aligning their coefficients, solve for the remaining variable, and then substitute it back to find the first variable.
To eliminate \(x\), we look for a common multiple of the coefficients of \(x\), which are 4 and 6. The least common multiple of 4 and 6 is 12.
First, multiply Equation 1 by 3:
\[3(4x + 3y) = 3(29) \implies 12x + 9y = 87 \quad \text{(Equation 3)}\]
Next, multiply Equation 2 by 2:
\[2(6x + 7y) = 2(41) \implies 12x + 14y = 82 \quad \text{(Equation 4)}\]
Now, subtract Equation 3 from Equation 4 to eliminate \(x\):
\[(12x + 14y) - (12x + 9y) = 82 - 87\]
\[5y = -5\]
Solving for \(y\), we divide both sides by 5:
\[y = -1\]
Now that we have the value of \(y = -1\), we substitute it into Equation 1 to solve for \(x\):
\[4x + 3(-1) = 29\]
\[4x - 3 = 29\]
Add 3 to both sides:
\[4x = 32\]
Divide by 4:
\[x = 8\]
Finally, verify the solution \(x=8\) and \(y=-1\) in Equation 2:
\[6(8) + 7(-1) = 48 - 7 = 41\]
The values satisfy both equations. Thus, the solution is \(x=8\) and \(y=-1\).
Systems of equations are used daily to optimize resources. For example, if you are planning a school event and need to buy two types of snacks, say popcorn bags and drink bottles, with a total budget and a specific number of items, you can use a system of equations to determine exactly how many of each you can afford. If \(x\) is the number of popcorn bags and \(y\) is the number of drink bottles, you might have equations like \(2x + 3y = 20\) representing cost, and \(x+y=8\) representing total items, allowing you to solve for the perfect combination.