- Analyze the given equations: \(y = -2x + b\) and \(4x + 2y = 8\).
- Convert the first equation into standard form: \(2x + y = b\).
- Simplify the second equation by dividing by \(2\): \(2x + y = 4\).
- Equate the constant terms to ensure the equations represent the same line: \(b = 4\).
- The final answer is \(\boxed{4}\).
Explanation
- Analyzing the System of Equations
To find the value of \(b\) that makes the system of equations have an infinite number of solutions, we must first analyze the two given equations:
- \(y = -2x + b\)
- \(4x + 2y = 8\)
A system of linear equations in two variables has an infinite number of solutions if the two equations are dependent, meaning they represent the exact same line on a 2D coordinate plane. This occurs when one equation is a scalar multiple of the other.
- Transforming the Equations
Let's rewrite the first equation, \(y = -2x + b\), in the standard form \(Ax + By = C\). By adding \(2x\) to both sides, we get:
\[2x + y = b\]
Next, let's simplify the second equation, \(4x + 2y = 8\). We can divide every term in this equation by \(2\) to isolate the coefficients:
\[\frac{4x}{2} + \frac{2y}{2} = \frac{8}{2}\]
\[2x + y = 4\]
- Solving for \(b\)
Now we have both equations in the same form:
- \(2x + y = b\)
- \(2x + y = 4\)
For these two equations to represent the same line, their coefficients for \(x\) and \(y\) must be proportional (which they already are, as both are \(1 \times (2x + y)\)), and their constant terms must be identical. Comparing the constants, we must have:
\[b = 4\]
When \(b = 4\), the two equations are identical, resulting in an infinite number of solutions.
Examples
In real-life scenarios, systems of equations can model constraints for production or resource allocation. For example, if two different production processes for the same product have the same total cost and yield, they are effectively the same process. Finding the point where these equations become identical helps in identifying redundant systems or ensuring that two different approaches are functionally equivalent, allowing for flexibility in choosing methods without changing the outcome.