- Substitute \(y = 4x + 4\) into the first equation \(2x + 2y = 13\).
- Simplify the resulting equation to find \(10x = 5\), which yields \(x = \frac{1}{2}\).
- Substitute \(x = \frac{1}{2}\) into \(y = 4x + 4\) to find \(y = 6\).
- The solution to the system is \(\boxed{(\frac{1}{2}, 6)}\).
Explanation
- Problem Analysis and Strategy
We are tasked with solving a system of two linear equations in two variables, \(x\) and \(y\), using the substitution method. The given equations are:
1) \(2x + 2y = 13\)
2) \(y = 4x + 4\)
The goal is to find the ordered pair \((x, y)\) that satisfies both equations. The substitution method involves replacing one variable in the first equation with an expression derived from the second equation. Since the second equation is already provided as \(y = 4x + 4\), we have a direct substitution path.
- Solving for x
We substitute the expression \(y = 4x + 4\) from the second equation into the first equation:
\[2x + 2(4x + 4) = 13\]
Next, we distribute the constant \(2\) into the parentheses:
\[2x + 8x + 8 = 13\]
Now, we combine the like terms involving \(x\):
\[10x + 8 = 13\]
To isolate the term with \(x\), we subtract \(8\) from both sides of the equation:
\[10x = 13 - 8\]
\[10x = 5\]
Finally, we solve for \(x\) by dividing both sides by \(10\):
\[x = \frac{5}{10} = \frac{1}{2}\]
- Finding the value of y
Now that we have the value of \(x = \frac{1}{2}\), we substitute this back into the second equation (\(y = 4x + 4\)) to find the corresponding value of \(y\):
\[y = 4\left(\frac{1}{2}\right) + 4\]
Perform the multiplication:
\[y = 2 + 4\]
\[y = 6\]
Thus, the solution to the system is \(x = \frac{1}{2}\) and \(y = 6\). The ordered pair is \((\frac{1}{2}, 6)\).
Examples
In business, substitution is commonly used to find the "break-even point" between two different scenarios. For example, if one supplier charges a fixed fee plus a cost per unit, and another charges a different fee structure, you can represent their costs as equations. By setting these equations equal or using substitution, you determine the exact number of units where both suppliers result in the same total cost, helping you make the most cost-effective decision.