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A linear equation has variables with a degree of 1, no products of variables, and forms a straight line when graphed. Examples include \( 2x + 3y = 6 \) and \( y = -4x + 7 \).
Solution
To determine which equation is linear, I can help you identify the characteristics of a linear equation. A **linear equation** typically has the following features:
1. **Degree of 1:** The highest power of the variable(s) is 1.
2. **No Products of Variables:** Variables are not multiplied together.
3. **Graph is a Straight Line:** When graphed on a coordinate plane, it forms a straight line.
A standard form of a linear equation in two variables is:
\[ y = mx + b \]
where:
- \( m \) is the slope,
- \( b \) is the y-intercept.
**Examples of Linear Equations:**
- \( 2x + 3y = 6 \)
- \( y = -4x + 7 \)
- \( 5 = 2x - y \)
**Non-Linear Equations:**
- \( y = x^2 + 3 \) (quadratic)
- \( y = \sqrt{x} + 2 \) (square root)
- \( y = \frac{1}{x} \) (rational)
If you provide the specific equations you're considering, I can help identify which one is linear!
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Simplify this solution Bonus Knowledge
A linear equation is one that can be represented in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For example, the equation \(2x + 3y = 6\) can be rearranged to this form, making it a linear equation. In contrast, equations that include exponents, products of variables, or absolute values typically are not linear! One common mistake is confusing linear equations with equations that look similar, like \(x^2 + y = 5\). Remember, if you see a variable raised to a power other than one, it’s likely not linear! A great tip is to always try to rearrange the equation into the slope-intercept form to confirm whether it’s linear.
