- Rewrite the equations in the standard form: \(3x - 4y = 3\) and \(9x + 7y = 8\).
- Use the elimination method by multiplying the first equation by 3 to match the x-coefficients.
- Solve for \(y\) by subtracting the equations, yielding \(y = -\frac{1}{19}\).
- Back-substitute \(y\) to find \(x = \frac{53}{57}\).
- The solution set is \(\boxed{\left(\frac{53}{57}, -\frac{1}{19}\right)}\).
Explanation
- Standardizing and Solving for y
The objective is to solve the system of linear equations:
\[\left\{\begin{array}{l} 3x = 4y + 3 \ 9x = 8 - 7y \end{array}\right.\]
First, we rewrite each equation in the standard form \(ax + by = c\):
- From the first equation, subtract \(4y\) from both sides to get \(3x - 4y = 3\).
- From the second equation, add \(7y\) to both sides to get \(9x + 7y = 8\).
We now have the system:
\[\left\{\begin{array}{l} 3x - 4y = 3 \quad \text{(Equation 1)} \ 9x + 7y = 8 \quad \text{(Equation 2)} \end{array}\right.\]
This system can be solved by the elimination method. By multiplying Equation 1 by 3, we can align the coefficients of \(x\):
\[3(3x - 4y) = 3(3) \implies 9x - 12y = 9 \quad \text{(Equation 3)}\]
Now, subtract Equation 3 from Equation 2 to eliminate \(x\):
\[(9x + 7y) - (9x - 12y) = 8 - 9\]
\[9x + 7y - 9x + 12y = -1\]
\[19y = -1\]
Dividing both sides by 19, we find the value of \(y\):
\[y = -\frac{1}{19}\]
- Solving for x and Concluding
With the value of \(y = -1/19\) known, we substitute this back into Equation 1 to solve for \(x\):
\[3x - 4\left(-\frac{1}{19}\right) = 3\]
\[3x + \frac{4}{19} = 3\]
Subtract \(\frac{4}{19}\) from both sides:
\[3x = 3 - \frac{4}{19}\]
Convert 3 to a fraction with a denominator of 19:
\[3 = \frac{3 \times 19}{19} = \frac{57}{19}\]
Thus,
\[3x = \frac{57}{19} - \frac{4}{19} = \frac{53}{19}\]
Dividing by 3 gives:
\[x = \frac{53}{19 \times 3} = \frac{53}{57}\]
The unique solution to the system is \((x, y) = \left(\frac{53}{57}, -\frac{1}{19}\right)\).
Examples
Systems of equations are fundamental in fields like economics and engineering to find equilibrium points. For instance, if you have two linear models representing supply and demand—where the price affects the quantity produced and purchased—finding the intersection (the solution) tells you the market equilibrium price and quantity. This method allows analysts to predict outcomes and make informed decisions based on varying constraints.