- Equate the two expressions for \(y\): \(\frac{1}{3}x - 14 = -x + 18\).
- Add \(x\) to both sides: \(\frac{4}{3}x - 14 = 18\).
- Add 14 to both sides: \(\frac{4}{3}x = 32\).
- Multiply by \(\frac{3}{4}\) to find the value: \(x = \boxed{24}\).
Explanation
- Analysis of the system of equations
We are given a system of two linear equations:
\[y = \frac{1}{3}x - 14\]
\[y = -x + 18\]
We need to find the value of \(x\) that satisfies both equations simultaneously. Since both expressions are equal to \(y\), we can set them equal to each other to solve for \(x\).
- Equating the expressions and solving for x
Setting the two expressions for \(y\) equal to each other gives:
\[\frac{1}{3}x - 14 = -x + 18\]
To isolate the terms involving \(x\), we add \(x\) to both sides of the equation:
\[\frac{1}{3}x + x - 14 = 18\]
\[\frac{4}{3}x - 14 = 18\]
Now, add 14 to both sides:
\[\frac{4}{3}x = 18 + 14\]
\[\frac{4}{3}x = 32\]
- Calculating the final value of x
To solve for \(x\), multiply both sides of the equation by the reciprocal of \(\frac{4}{3}\), which is \(\frac{3}{4}\):
\[x = 32 \times \frac{3}{4}\]
\[x = \frac{32 \times 3}{4}\]
\[x = 8 \times 3 = 24\]
Thus, the value of \(x\) that satisfies the system of equations is 24.
Examples
Systems of linear equations are used in many real-life situations, such as determining the break-even point for a business. For example, if you sell handmade crafts, you might have a cost equation \(y = mx + b\) representing your expenses and a revenue equation \(y = rx\) representing your income. The point where these two lines intersect, the solution \((x, y)\), tells you exactly how many items you need to sell for your revenue to cover your costs. In this case, solving for \(x\) helps you identify the exact number of units that leads to zero profit (or loss).