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Chapter 9: Problem 26

In Exercises \(25-30,\) find the slope and \(y\) -intercept of the line by writing the cquation in slope-intercept form. \(6 x+2 y-10=0\)

Short Answer

Expert verified
The slope is \( -3\) and the y-intercept is \(5\).

Step by step solution

01

Write the equation in standard form

The original equation is given as: \[6x + 2y - 10 = 0\]
02

Isolate the y-term

Move all terms involving \(x\) and the constant term to the right-hand side of the equation: \[2y = -6x + 10\]
03

Solve for y

Divide every term on both sides by 2 to isolate \(y\): \[ y = -3x + 5\]
04

Identify the slope and y-intercept

The equation is now in slope-intercept form (y = mx + b), where \(m\) is the slope and \(b\) is the y-intercept. So, \(m = -3\) and \(b = 5\).

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Key Concepts

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

standard form equation
A standard form equation of a line is written as Ax + By = C where A, B, and C are real numbers, and both A and B are not zero simultaneously. In this problem, the given equation is 6x + 2y - 10 = 0. It is in a standard form that can make it easier to manipulate terms and solve for y.
isolate y-term
The next step is to isolate the y-term. Begin by moving the terms involving x and any constants to the other side of the equation.
In the given equation 6x + 2y - 10 = 0, we can subtract 6x from both sides and add 10 to both sides, resulting in 2y = -6x + 10.
By doing this, we make it easier to solve for y in the next step.
solve for y
To solve for y, you need to isolate y completely. In the equation 2y = -6x + 10, you should divide each term by 2:
Which gives us the equation y = -3x + 5. This is the slope-intercept form of the equation, y = mx + b, where 'm' and 'b' are constants representing the slope and y-intercept, respectively.
identify slope and y-intercept
Now that the equation is in the slope-intercept form y = -3x + 5, it's time to identify the slope and y-intercept. In this form, the slope is the coefficient of x, which is -3, and the y-intercept is the constant term, which is 5.
Here:
  • Slope (m) = -3
  • y-intercept (b) = 5
These tell us that the line crosses the y-axis at (0, 5) and for each unit increase in x, the value of y decreases by 3.

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Most popular questions from this chapter

In Exercises \(33-38\), determine whether the lines with the given equations are parallel, perpendicular, or neither. \(5 x-3 y+8=0\) \(3 x+5 y-7=0\)

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. Through \(\left(2, \frac{1}{2}\right)\) and \((-3,8)\)

Find the distance between the points given. $$(3,2) \text { and }(-2,-10)$$

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. Through \((-2,-4)\) with slope \(-3\)

Determine the intercepts and graph each linear equation. $$3 x+2 y=6$$
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