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Chapter 3: Problem 4

Write the equation \(y+2=-3(x-4)\) in slope-intercept form.

Short Answer

Expert verified
y = -3x + 10

Step by step solution

01

- Identify the given equation

The given equation is in point-slope form: y + 2 = -3(x - 4)Our goal is to convert it to slope-intercept form y = mx + b.
02

- Expand the right side of the equation

Expand the right side to distribute -3:y + 2 = -3x + 12
03

- Isolate y

Subtract 2 from both sides to isolate y:y = -3x + 12 - 2Simplify the equation:y = -3x + 10

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form
The point-slope form of a linear equation is quite useful starting point, especially when you have a point and a slope. This form highlights the relationship between the x and y coordinates of the points on the line. The general form is: y - y_1 = m(x - x_1) where m is the slope and (x_1, y_1) is a specific point on the line. For example, given y + 2 = -3(x - 4), the slope (m) is -3, and the point (x_1, y_1) would be (4, -2). Using this form makes it easier to transition to other forms, like the slope-intercept form.
Distributive Property
The distributive property is crucial when you need to simplify or expand equations. It states that to distribute a multiplication over an addition or subtraction within parentheses, you multiply each term inside the parentheses by the term outside. In mathematical terms: a(b + c) = ab + ac Applying it to our problem, we have y + 2 = -3(x - 4). To apply the distributive property, you multiply -3 by both x and -4. This results in: y + 2 = -3x + 12. Now, the equation is ready for further simplification towards the slope-intercept form.
Isolating Variables
One important step in converting equations to different forms is isolating the variable you want to solve for. Here, we need to isolate y. Starting from y + 2 = -3x + 12, you subtract 2 from both sides to remove the constant on the left side: y + 2 - 2 = -3x + 12 - 2 which simplifies to: y = -3x + 10. Now, the equation is in the slope-intercept form: y = mx + b where m is the slope, and b is the y-intercept.

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Most popular questions from this chapter

Find the midpoint of each segment with the given endpoints. $$ (-10,4) \text { and }(7,1) $$

For each situation, (a) write an equation in the form \(y=m x+b,(b)\) find and interpret the ordered pair associated with the equation for \(x=5,\) and \((c)\) answer the question. A membership in the Midwest Athletic Club costs \(\$ 99,\) plus \(\$ 41\) per month. (Source: Midwest Athletic Club.) Let \(x\) represent the number of months and \(y\) represent the cost. How much does the first year's membership cost?

Use your knowledge of the slopes of parallel and perpendicular lines. Is the figure with vertices at \((-11,-5),(-2,-19),(12,-10),\) and \((3,4)\) a parallelogram? Is it a rectangle? (Hint: A rectangle is a parallelogram with a right angle.)

Find the midpoint of each segment with the given endpoints. $$ (4,-3) \text { and }(-1,3) $$

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. See Examples \(1-3\). $$ f(-x) $$
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