Find the gradient and the y-intercept of the line 2x + 10y = 20.
Understand the Problem
The question asks for the gradient and y-intercept of the given linear equation, which involves manipulating the equation into slope-intercept form (y = mx + b) where m is the gradient and b is the y-intercept.
Answer
The gradient is $-\frac{1}{5}$ and the y-intercept is $2$.
Answer for screen readers
The gradient is $-\frac{1}{5}$ and the y-intercept is $2$.
Steps to Solve
- Rearrange the equation to slope-intercept form
Start with the given equation: $$ 2x + 10y = 20 $$
To express it in slope-intercept form ($y = mx + b$), isolate $y$. First, subtract $2x$ from both sides:
$$ 10y = -2x + 20 $$
- Solve for y
Now, divide all terms by 10 to solve for $y$:
$$ y = -\frac{2}{10}x + \frac{20}{10} $$
This simplifies to:
$$ y = -\frac{1}{5}x + 2 $$
- Identify the gradient and y-intercept
From the slope-intercept form $y = mx + b$, we identify:
- The gradient ($m$) is $-\frac{1}{5}$
- The y-intercept ($b$) is $2$
The gradient is $-\frac{1}{5}$ and the y-intercept is $2$.
More Information
The gradient indicates that for every 5 units the line moves down, it moves 1 unit to the right. The y-intercept represents the point where the line crosses the y-axis at (0, 2).
Tips
- Misinterpreting the coefficients: Ensure you correctly identify the coefficients when converting to slope-intercept form.
- Forgetting to simplify: When dividing terms, always simplify fractions to their lowest terms.
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