- Analyze the equation: \(-10x + 11(y + x) = 10\).
- Expand the expression using the distributive property: \(-10x + 11y + 11x = 10\).
- Simplify by combining like terms to get the standard form: \(x + 11y = 10\).
- Identify the equation as linear, with the standard form: \(\boxed{x + 11y = 10}\).
Explanation
- Problem Analysis
A linear equation in two variables \(x\) and \(y\) is an equation that can be expressed in the standard form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and at least one of \(a\) or \(b\) is non-zero. The given equation is \(-10x + 11(y + x) = 10\). To determine if it is linear, we need to expand and simplify it to see if it fits the standard form.
- Simplifying the Equation
We begin by applying the distributive property to the term \(11(y + x)\):
\[-10x + 11y + 11x = 10\]
Next, we combine the like terms involving \(x\). The coefficients of \(x\) are \(-10\) and \(11\). Adding them gives:
\[(-10 + 11)x + 11y = 10\]
\[1x + 11y = 10\]
This simplifies to:
\[x + 11y = 10\]
This expression is now in the standard form \(ax + by = c\), where \(a=1\), \(b=11\), and \(c=10\).
- Conclusion
Since the simplified equation \(x + 11y = 10\) satisfies the definition of a linear equation (it is a polynomial equation of degree 1 in two variables), the equation is linear. Its standard form is \(x + 11y = 10\).
Examples
Linear equations are fundamental in modeling real-world situations with constant rates of change. For instance, if you are buying items for a school project, say notebooks at \$1 each and pens at \$11 each, and you have a total budget of \$10, the equation \(x + 11y = 10\) describes all possible combinations of notebooks (\(x\)) and pens (\(y\)) you can purchase. This helps in budgeting and resource management.