- Identify the x-coordinate (\(-1\)) and y-coordinate (\(4\)) from the given point.
- Substitute \(x = -1\) and \(y = 4\) into the equation \(3x + 8y = 29\).
- The resulting equation is \(3 \cdot (-1) + 8 \cdot (4) = 29\).
- Final answer: \(\boxed{3 \cdot(-1)+8 \cdot(4)=29}\).
Explanation
- Analyzing the Problem and Data
To determine if the ordered pair \((-1, 4)\) is a solution to the linear equation \(3x + 8y = 29\), we need to substitute the given coordinates into the equation. An ordered pair \((x, y)\) consists of an x-coordinate and a y-coordinate. In this case, \(x = -1\) and \(y = 4\). Replacing these values into the original expression \(3x + 8y\) allows us to see if the equation holds true.
- Substitution and Verification
The linear equation is given by \(3x + 8y = 29\). We are testing the point \((-1, 4)\). Substituting \(x = -1\) into the term \(3x\), we get \(3 \times (-1)\). Substituting \(y = 4\) into the term \(8y\), we get \(8 \times 4\). Adding these together according to the original equation, we obtain the expression \(3 \times (-1) + 8 \times 4\). Comparing this to the given options, we find the correct form.
- Conclusion
The equation that correctly represents the substitution of the point \((-1, 4)\) into \(3x + 8y = 29\) is \(3 \cdot (-1) + 8 \cdot (4) = 29\). This is the fourth option presented in the question. Evaluating it, we get \(-3 + 32 = 29\), which is correct. Therefore, the point \((-1, 4)\) is indeed a solution.
Examples
In real-life situations, such as checking if a specific budget fits within a cost model, you often need to verify if certain variables satisfy an equation. For example, if the total cost of a shopping trip is modelled by \(3x + 8y = 29\), where \(x\) is the number of items at \(\$3\) and \(y\) is the number of items at \(\$8\), you would plug in your item counts to verify the total cost. Knowing how to substitute values correctly ensures your calculations and financial planning remain accurate.