$y-1=\frac{1}{2}(x+2)$
Answer (2)
Rewrite the given equation y − 1 = 2 1 ( x + 2 ) in slope-intercept form.
Distribute and isolate y to get y = 2 1 x + 2 .
Convert the slope-intercept form to standard form by rearranging terms: x − 2 y = − 4 .
The equation is now in both slope-intercept and standard forms: y = 2 1 x + 2 and x − 2 y = − 4 , respectively. y = 2 1 x + 2
Explanation
Understanding the Equation We are given the equation of a line: y − 1 = 2 1 ( x + 2 ) . Our goal is to rewrite this equation in different forms to better understand its properties, such as the slope and intercepts.
Converting to Slope-Intercept Form First, let's rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Starting with the given equation:
y − 1 = 2 1 ( x + 2 )
Distribute the 2 1 on the right side:
y − 1 = 2 1 x + 1
Add 1 to both sides to isolate y :
y = 2 1 x + 1 + 1
y = 2 1 x + 2
So, the slope-intercept form of the equation is y = 2 1 x + 2 . From this, we can see that the slope m = 2 1 and the y-intercept b = 2 .
Converting to Standard Form Next, let's rewrite the equation in standard form, which is A x + B y = C , where A , B , and C are integers, and A is usually non-negative. Starting from the slope-intercept form we found:
y = 2 1 x + 2
Subtract 2 1 x from both sides:
− 2 1 x + y = 2
To eliminate the fraction, multiply the entire equation by 2:
2 ( − 2 1 x + y ) = 2 ( 2 )
− x + 2 y = 4
Multiply by -1 to make the coefficient of x positive:
x − 2 y = − 4
So, the standard form of the equation is x − 2 y = − 4 .
Final Answer In summary, we have analyzed the given equation of a line and rewritten it in two common forms: slope-intercept form and standard form. The slope-intercept form is y = 2 1 x + 2 , which tells us the slope is 2 1 and the y-intercept is 2. The standard form is x − 2 y = − 4 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, in economics, you might use a linear equation to model the relationship between the price of a product and the quantity demanded. The slope would represent how much the quantity demanded changes for each unit change in price. Similarly, in physics, you could use a linear equation to describe the motion of an object moving at a constant velocity, where the slope represents the velocity and the y-intercept represents the initial position.
The equation y − 1 = 2 1 ( x + 2 ) can be rewritten in slope-intercept form as y = 2 1 x + 2 and in standard form as x − 2 y = − 4 . The slope is 2 1 and the y-intercept is 2. Both forms help describe the linear relationship represented by the equation.
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