Problem 55 Find the slope and the \(y\) -in... [FREE SOLUTION] (2024)

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Chapter 1: Problem 55

Find the slope and the \(y\) -intercept of the line with the given equation. $$3 x+2 y=10$$

Short Answer

Expert verified

The slope is \(-\frac{3}{2}\), and the \(y \, -intercept\) is \5\.

Step by step solution


- Write the equation in slope-intercept form

The slope-intercept form of a line is given by \(y = mx + c\), where \m\ is the slope and \c\ is the \(y \, -intercept\). Start by isolating \(y\) in the given equation \(3x + 2y = 10\).


- Subtract 3x from both sides

To isolate \(y\), subtract \(3x\) from both sides: \(2y = -3x + 10\)


- Divide by 2

Next, divide every term by \(2\) to solve for \(y\): \[y = -\frac{3}{2}x + 5\]


- Identify the slope and \(y\)-intercept

In the equation \[y = -\frac{3}{2}x + 5\], the slope \(m\) is \(-\frac{3}{2}\) and the \(y \, -intercept\) \(c\) is \(5\).

Key Concepts

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

linear equations

Linear equations are equations involving two variables, often written in the form of \(ax + by = c\). These equations create a straight line when plotted on a graph. For instance, the equation \(3x + 2y = 10\) is a linear equation. The main goal when working with linear equations is to explore the relationship between the variables \(x\) and \(y\). In math, it's common to rearrange these equations to reveal more information about the line's properties such as its slope and \(y\)-intercept.

slope-intercept form

The slope-intercept form is a specific arrangement of a linear equation. You'll see it written as \(y = mx + c\). This form is quite handy because it immediately reveals two important characteristics about the line: the slope (\(m\)) and the \(y\)-intercept (\(c\)).

  • The slope, denoted by \(m\), tells us the steepness and direction of the line. If \(m\) is positive, the line rises; if \(m\) is negative, the line falls.
  • The \(y\)-intercept, denoted by \(c\), is the point where the line crosses the \(y\)-axis. It's what \(y\) equals when \(x\) is \(0\).

To convert any linear equation into this form, you'll need to isolate \(y\), a process we call 'isolation of variables'.

isolation of variables

Isolating variables means rearranging an equation to solve for one variable in terms of others. This is key in simplifying linear equations and finding specific solutions. Let's revisit our example, \(3x + 2y = 10\):

  • First, we subtract \(3x\) from both sides to get: \(2y = -3x + 10\).
  • Next, we divide each term by \(2\) to isolate \(y\): \(y = -\frac{3}{2}x + 5\).

Now, the equation is in slope-intercept form. This shows the slope (\(m\)) is \(-\frac{3}{2}\) and the \(y\)-intercept (\(c\)) is \(5\). Understanding how to isolate variables will help you solve many different types of equations and understand the relationships between variables more clearly.

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Problem 55 Find the slope and the \(y\) -in... [FREE SOLUTION] (3)

Most popular questions from this chapter

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Problem 55 Find the slope and the \(y\) -in... [FREE SOLUTION] (2024)


How do you find the slope and Y value? ›

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 the y-intercept for the graph of this equation y − 3x − 2? ›

For the given equation, the slope is -3, which means for every 1 unit increase in the x-coordinate, the y-coordinate will decrease by 3 units. The y-intercept is -2, which represents the point where the line crosses the y-axis.

What is the slope for 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 and y-intercept of the function? ›

Summary. The slope-intercept form of a line is: y=mx+b where m is the slope and b is the y-intercept. The y-intercept is always where the line intersects the y-axis, and will always appear as (0,b) in coordinate form.

How to find y-intercept? ›

To find y-intercept: set x = 0 and solve for y. The point will be (0, y).

How do you answer slope? ›

The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

How to find the y-intercept with two points? ›

Use the slope and one of the points to solve for the y-intercept (b). One of your points can replace the x and y, and the slope you just calculated replaces the m of your equation y = mx + b. Then b is the only variable left. Use the tools you know for solving for a variable to solve for b.

Can you find the y-intercept with just the slope? ›

use the slope-intercept formula: y=m*x+c. here the slope is m, and use the point (0,y), in the formula, y=m*(0)+c, here the y-intercept is the point (0,c). since the slope is m*(0) we are left with y=c, therefore the y-intercept is the y-value of the slope, where the slope is y/0.

What is the Y formula for slope? ›

y = mx + b is the slope-intercept form of the equation of a straight line. In the equation y = mx + b, m is the slope of the line and b is the intercept. x and y represent the distance of the line from the x-axis and y-axis, respectively. The value of b is equal to y when x = 0, and m shows how steep the line is.

What is the formula for the slope value? ›

tan θ = Δy/Δx

So, tan θ to be the slope of a line. Generally, the slope of a line gives the measure of its steepness and direction. The slope of a straight line between two points says (x1,y1) and (x2,y2) can be easily determined by finding the difference between the coordinates of the points.

What is the slope of y =- 6x 3? ›

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

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