Solve the simultaneous equations.
$\begin{array}{r}
3 x+4 y=9 \\
x+2 y=5
\end{array}$
We are given a system of two linear equations with two variables \(x\) and \(y\):
1) \(3x + 4y = 9\)
2) \(x + 2y = 5\)
Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. To do this, we will use the method of elimination.
To eliminate one variable, we look at the coefficients. Equation (2) has a coefficient of 2 for \(y\), while Equation (1) has a coefficient of 4 for \(y\). By multiplying Equation (2) by 2, the coefficient of \(y\) will match Equation (1):
\(2 \times (x + 2y) = 2 \times 5\)
\(2x + 4y = 10\)
Now we have:
1) \(3x + 4y = 9\)
3) \(2x + 4y = 10\)
Subtracting Equation (3) from Equation (1), we eliminate \(y\):
\((3x + 4y) - (2x + 4y) = 9 - 10\)
\(3x - 2x + 4y - 4y = -1\)
\(x = -1\)
Now that we have found \(x = -1\), we substitute this value into the original Equation (2) to solve for \(y\):
\((-1) + 2y = 5\)
Adding 1 to both sides:
\(2y = 6\)
Dividing by 2:
\(y = 3\)
Finally, we can verify this solution by checking it in Equation (1):
\(3(-1) + 4(3) = -3 + 12 = 9\)
The values \(x = -1\) and \(y = 3\) satisfy both equations.
Systems of linear equations are widely used in economics and business to find the equilibrium point where supply equals demand. For instance, if you are running a business, you might have one linear equation representing your costs and another representing your revenue. The point where the two lines intersect, known as the break-even point, is the solution to the system of equations, telling you exactly how many items you need to sell to start making a profit. This algebraic method of finding the intersection of two lines provides a foundation for more complex modeling and decision-making.