Problem 24 Find the slope and \(y\) -interc... [FREE SOLUTION] (2024)

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Chapter 0: Problem 24

Find the slope and \(y\) -intercept. $$3 x-3 y+6=0$$

Short Answer

Expert verified

Slope: 1, y-intercept: 2.

Step by step solution


- Rewrite the equation in slope-intercept form

Start by rewriting the given equation in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. The given equation is \(3x - 3y + 6 = 0\).


- Solve for y

Divide every term by \(-3\) to solve for \(y\): \(\frac{-3y}{-3} = \frac{-3x}{-3} + \frac{-6}{-3}\). Simplifying this, we get \(y = x + 2\).


- Identify the slope and y-intercept

Now that we have the equation in slope-intercept form \(y = mx + b\), we can identify the slope (\text{m}\text{ in the equation) and \(y\)-intercept (\text{b}). Here, \(m = 1\) and \(b = 2\).

Key Concepts

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


In the context of linear equations, the slope is a measure of how steep a line is. It tells us how much the y-value (vertical) changes for a one-unit change in the x-value (horizontal).
The slope is often represented by the letter 'm'. In the slope-intercept form of a linear equation, which is written as \( y = mx + b \), 'm' is the slope.

To find the slope, we look at the coefficient of the x term. Slope can also be calculated by the rise-over-run formula, which is \[ m = \frac{\Delta y}{\Delta x} \]
This formula means the change in y divided by the change in x. In our example, after converting the equation to slope-intercept form \( y = x + 2 \). Here, the slope \( m = 1 \).

Key takeaways about slope:

  • It measures the steepness of the line.
  • Calculated as the ratio of the change in y to the change in x.
  • A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.

The y-intercept is the point where the line crosses the y-axis. In other words, it's the value of y when x is zero. It's represented by 'b' in the slope-intercept form of the equation \( y = mx + b \).
To find the y-intercept, we simply look at the constant term in the equation. In our example, the equation \( y = x + 2 \) shows a y-intercept of 2. This means that when x is 0, y is 2.

Understanding the y-intercept can be very helpful:

  • It shows where the line starts on the y-axis.
  • Helps in graphing the line quickly.
  • Gives a point of reference for the line.

Remember, the y-intercept is always found by setting x to 0 in the equation and solving for y.

linear equations

Linear equations are mathematical statements that create a straight line when plotted on a graph. They usually come in the form \( ax + by = c \) or can be rewritten as \( y = mx + b \) (slope-intercept form).
The main feature of linear equations is that the relationship between the variables x and y is constant, meaning the graph will always be a straight line.

For example, in the equation \( 3x - 3y + 6 = 0 \), we can isolate y to get the slope-intercept form \( y = x + 2 \). This transformation makes it easier to understand the graph of the line.

Key points to remember about linear equations:

  • They graph as straight lines.
  • The slope-intercept form \( y = mx + b \) is very useful for understanding the slope and y-intercept.
  • Linear equations represent linear relationships between variables.

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is used to express and solve equations and inequalities.
In our example, algebra helps us transform the given equation \( 3x - 3y + 6 = 0 \) into the slope-intercept form \( y = x + 2 \).

Understanding some basic concepts of algebra is essential:

  • Variables: Symbols like x and y that represent numbers.
  • Equations: Mathematical statements that show the equality of two expressions.
  • Operations: Basic arithmetic (addition, subtraction, multiplication, division), but applied to variables and constants.

With algebra, we can manipulate equations and inequalities to find solutions to various problems. It's a fundamental tool for advanced mathematics and real-world problem-solving.

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Problem 24 Find the slope and \(y\) -interc... [FREE SOLUTION] (3)

Most popular questions from this chapter

For \(f(x)=x^{2}+4 x,\) find \(\frac{f(x+h)-f(x)}{h}.\)Find the domain of each function given below. $$f(x)=\sqrt{2 x}$$The stopping distance (at some fixed speed) of regular tires on glare ice isgiven by a linear function of the air temperature \(F\) \(D(F)=2 F+115\) where\(D(F)\) is the stopping distance, in feet, when the air temperature is \(F\), indegrees Fahrenheit. a) Find \(D\left(0^{\circ}\right), D\left(-20^{\circ}\right),D\left(10^{\circ}\right),\) and \(D\left(32^{\circ}\right).\) b) Explain why the domain should be restricted to the interval\(\left[-57.5^{\circ}, 32^{\circ}\right].\)(See Exercise 68.) The Video Wizard buys a new computer system for \(\$ 60,000\)and projects that its book value will be \(\$ 2000\) after 5 yr. Using straight-line depreciation, find the book value after 3 yr.Find the domain of each function given below. $$f(x)=\frac{2}{x+3}$$
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Problem 24 Find the slope and \(y\) -interc... [FREE SOLUTION] (2024)


How to solve for slope and y-intercept? ›

The equation of the line is written in the slope-intercept form, which is: y = mx + b, where m represents the slope and b represents the y-intercept. In our equation, y = 6x + 2, we see that the slope of the line is 6.

What is the slope and y-intercept of this line y =- 6x 2? ›

The slope of the line is -6 and the y-intercept is 2.

What is the slope of 14x 4y 24? ›

Expert-Verified Answer

The slope and y-intercept of the line will be -7/2 and 6.

What is the slope of the line whose equation is 5y 6x 2 0? ›

Summary: The slope of the line whose equation is 5y + 6x - 2 = 0 is -6/5.

How to solve for slope? ›

The slope, or steepness, of a line is found by dividing the vertical change (rise) by the horizontal change (run). The formula is slope =(y₂ - y₁)/(x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the coordinates of two points on the line.

What is the slope of y? ›

In an equation in slope-intercept form (y=mx+b) the slope is m and the y-intercept is b. We can also rewrite certain equations to look more like slope-intercept form. For example, y=x can be rewritten as y=1x+0, so its slope is 1 and its y-intercept is 0.

What is the slope of \[ y = 4 2x \]? ›

Using the slope-intercept form, the slope is −2 .

What is the slope of y =- 6x 8? ›

The Slope Is: m=−6.

What is the slope intercept of y =- 2? ›

Explanation: y=−2 is a flat line crossing the point at (0,−2) therefore the gradient is zero and the the y intercept is -2.

What is the slope of 8x 6y =- 24? ›

8x-6y=-24 Geometric figure: Straight Line Slope = 2.667/2.000 = 1.333 x-intercept = -12/4 = -3 y-intercept = 12/3 = 4 Rearrange: Rearrange the equation by subtracting what is to the ...

What is the slope of 4x 7y 6? ›

Using the slope-intercept form, the slope is −47 .

What is the slope of 8x 4y =- 12? ›

Add 8x to both sides: -4y= 8x- 12. Divide both sides by -4: y= -2x+ 3. The slope of the line is -2.

What is the slope of the line whose equation is y = 0? ›

Because the equation y=0 has a slope of 0, the resulting graph must be a straight horizontal line. This is true for all equations with a slope of 0.

What is the slope of the graph of y 6x 11? ›

Using the slope-intercept form, the slope is 6 .

How do you find the slope of Y 6x 4? ›

Explanation: Since the equation of a line in slope-intercept form is y=mx+b , with m representing the slope and b representing the y-intercept, the slope would be 6 and the y-intercept 4.

What is the formula for the slope intercept? ›

The slope intercept form in math is one of the forms used to calculate the equation of a straight line, given the slope of the line and intercept it forms with the y-axis. The slope intercept form is given as, y = mx + b, where 'm' is the slope of the straight line and 'b' is the y-intercept.

How to find slope and y-intercept from standard form? ›

Or we can convert the standard form equation to the slope intercept equation y=mx+ b; using this equation, the value m=slope and b=y intercept.

What is the slope of y =- 6x 3? ›

Using the slope-intercept form, the slope is −6 .

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