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Chapter 7: Problem 58

Write an equation of the line in slope-intercept form that passes through the two points, or passes through the point and has the given slope. $$(-2,4), m=3$$

Short Answer

Expert verified
The equation of the line in slope-intercept form that passes through the points (-2,4) and has the given slope 3 is \(y = 3x + 10\).

Step by step solution

01

Identify the given slope and point

In the problem, it is given that the slope \(m\) is 3 and the line passes through the point (-2,4). So, \(m = 3\) and \(x_1 = -2\) and \(y_1 = 4\).
02

Substitute the known values into the formula to solve for \(b\)

The formula to find the y-intercept \(b\) when you have the slope and one point \(x_1, y_1\) is: \(b = y_1 - m*x_1\). Substituting the given values, we get: \(b = 4 - 3*(-2) = 4 + 6 = 10\).
03

Write the final slope-intercept form of the line

Upon substituting the value of slope \(m = 3\) and y-intercept \(b = 10\) into the slope-intercept form equation \(y = mx + b\), the equation of the line becomes \(y = 3x + 10\).

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Key Concepts

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

Writing Linear Equations
Understanding how to write linear equations is fundamental in algebra, especially in coordinate geometry. The most common form is the slope-intercept form, which is expressed as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept.

To write an equation, you need to know at least the slope and one point, or two points through which the line passes. Once you have these, you plug the values into the slope-intercept form. If only a point and the slope are given, such as the point (-2,4) with a slope of 3, you directly substitute the slope in place of \(m\), and use the point to solve for \(b\), the y-intercept, as shown in the provided solution steps.
Slope of a Line
The slope of a line is a measure of its steepness or inclination and is a crucial concept in coordinate geometry. It is indicated by the letter \(m\) and calculated as the 'rise over run', or the change in y over the change in x between two points on the line.

If you have two points, \(x_1, y_1\) and \(x_2, y_2\), the slope is found using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). A positive slope means the line is rising as it moves from left to right, a negative slope means it's falling, and a slope of zero indicates a horizontal line. In the given example, the slope is 3, meaning for every unit the line moves horizontally, it rises by 3 units vertically.
Y-Intercept
The y-intercept of a line is the point where it crosses the y-axis, and it's always expressed as an ordered pair with zero as the x-coordinate, like \(0, b\).

In the slope-intercept form equation \(y = mx + b\), \(b\) denotes the y-intercept. To find \(b\) when you have a slope and a point, use the formula \(b = y_1 - m*x_1\), as done in the second step of our solution. It's pivotal because it helps to anchor the line in place on a graph. In our example, after working out the math, we find the y-intercept is 10, giving us the coordinate \(0, 10\) where our line intercepts the y-axis.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, lets us describe geometric shapes algebraically using coordinates and equations. It's a way of combining algebra and geometry to solve problems involving lines, angles, and curves.

In the context of linear equations, coordinate geometry helps you visualize the slope and y-intercept as parts of a line in a two-dimensional space. With the equation of the line \(y = 3x + 10\), you can plot it on a graph using the slope and y-intercept. This method is used to solve various real-world problems, like predicting trends or determining the distance between two points.

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Most popular questions from this chapter

Evaluate the expression. \(\left(3^{3}-20\right)^{2}\)

Use the following information. You own a bottle recycling center that receives bottles that are either sorted by color or unsorted. To sort and recycle all of the bottles, you can use up to 4200 hours of human labor and up to 2400 hours of machine time. The system below represents the number of hours your center spends sorting and recycling bottles where \(x\) is the number of tons of unsorted bottles and \(y\) is the number of tons of sorted bottles. \(4 x+y \leq 4200\) \(2 x+y \leq 2400\) \(x \geq 0, y \geq 0\) Graph the system of linear inequalities.

Use the following information. You own a bottle recycling center that receives bottles that are either sorted by color or unsorted. To sort and recycle all of the bottles, you can use up to 4200 hours of human labor and up to 2400 hours of machine time. The system below represents the number of hours your center spends sorting and recycling bottles where \(x\) is the number of tons of unsorted bottles and \(y\) is the number of tons of sorted bottles. \(4 x+y \leq 4200\) \(2 x+y \leq 2400\) \(x \geq 0, y \geq 0\) a. Find the vertices of your graph. b. You will earn 30 dollar per ton for bottles that are sorted by color and earn 10 dollar per ton for unsorted bottles. Let \(P\) be the maximum profit. Substitute the ordered pairs from part (a) into the following equation and solve. $$ P=10 x+30 y $$ c. Assuming that the maximum profit occurs at one of the four vertices, what is the maximum profit?

Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form. $$ (-1,5), m=\frac{2}{3} $$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&4 x+4 y=-8\\\&2 x+2 y=-4\end{aligned} $$
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