# Introduktion til hældningen-skæringspunkt-form

Learn about the slope-intercept form of two-variable linear equations, and how to interpret it to find the slope and y-intercept of their line.

#### What you will learn in this lesson

• What is the slope-intercept form of two-variable linear equations
• How to find the slope and the y-intercept of a line from its slope-intercept equation
• How to find the equation of a line given its slope and y-intercept

Slope-intercept is a specific form of linear equations. It has the following general structure. Drum roll ...
y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE
Here, start color maroonC, m, end color maroonC and start color greenE, b, end color greenE can be any two real numbers. For example, these are linear equations in slope-intercept form:
• y, equals, 2, x, plus, 1
• y, equals, minus, 3, x, plus, 2, point, 7
• y, equals, 10, minus, 100, x
It's true that this equation is different from the previous ones because the constant term—i.e., the plain number—comes before the x-term.
However, because addition is commutative, this is essentially the same form. You can say that this equation is of the form y, equals, start color greenE, b, end color greenE, plus, start color maroonC, m, end color maroonC, x instead of y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE, but these mean the same thing.
On the other hand, these linear equations are not in slope-intercept form:
• 2, x, plus, 3, y, equals, 5
• y, minus, 3, equals, 2, left parenthesis, x, minus, 1, right parenthesis
• x, equals, 4, y, minus, 7
Slope-intercept is the most prominent form of linear equations. Let's dig deeper to learn why this is so.

# Koefficienterne i hældning-skæringspunkts-form

Besides being neat and simplified, slope-intercept form's advantage is that it gives two main features of the line it represents:
• Hældningen af linjen er start color maroonC, m, end color maroonC.
• The y-coordinate of the y-intercept is start color greenE, b, end color greenE. In other words, the line's y-intercept is at left parenthesis, 0, comma, start color greenE, b, end color greenE, right parenthesis.
For example, the line y, equals, start color maroonC, 2, end color maroonC, x, start color greenE, plus, 1, end color greenE has a slope of start color maroonC, 2, end color maroonC and a y-intercept at left parenthesis, 0, comma, start color greenE, 1, end color greenE, right parenthesis:
The fact that this form gives the slope and the y-intercept is the reason why it is called slope-intercept in the first place!

1) What is the slope of the line represented by y, equals, 5, x, minus, 7?space

Remember that in the general form y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE, the slope is given by start color maroonC, m, end color maroonC.
Therefore, y, equals, start color maroonC, 5, end color maroonC, x, start color greenE, minus, 7, end color greenE has a slope of start color maroonC, 5, end color maroonC.
2) What is the slope of the line represented by y, equals, x, plus, 9?space

Remember that in the general form y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE, the slope is given by start color maroonC, m, end color maroonC.
Furthermore, if x appears by itself, it's the same thing as x having a coefficient of one. In other words, the given equation can be written as y, equals, start color maroonC, 1, end color maroonC, x, start color greenE, plus, 9, end color greenE.
Therefore, y, equals, start color maroonC, 1, end color maroonC, x, start color greenE, plus, 9, end color greenE has a slope of start color maroonC, 1, end color maroonC.
3) What is the y-intercept of the line represented by y, equals, minus, 6, x, minus, 11?

Remember that in the general form y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE, the y-intercept is given by start color greenE, b, end color greenE.
Furthermore, at the y-intercept, x is always equal to zero.
Therefore, y, equals, start color maroonC, minus, 6, end color maroonC, x, start color greenE, minus, 11, end color greenE has a y-intercept at left parenthesis, 0, comma, start color greenE, minus, 11, end color greenE, right parenthesis.
4) What is the y-intercept of the line represented by y, equals, 4, x?

Remember that in the general form y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE, the y-intercept is given by start color greenE, b, end color greenE.
Furthermore, if start color greenE, b, end color greenE doesn't appear, it's the same thing as start color greenE, b, end color greenE being zero. In other words, the given equation can be written as y, equals, start color maroonC, 4, end color maroonC, x, plus, start color greenE, 0, end color greenE.
Therefore, y, equals, start color maroonC, 4, end color maroonC, x, plus, start color greenE, 0, end color greenE has a y-intercept at left parenthesis, 0, comma, start color greenE, 0, end color greenE, right parenthesis.
5) What is the slope of the line represented by y, equals, 1, minus, 8, x?space

Remember that in the general form y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE, the slope is given by start color maroonC, m, end color maroonC.
Therefore, y, equals, start color greenE, 1, end color greenE, start color maroonC, minus, 8, end color maroonC, x has a slope of start color maroonC, minus, 8, end color maroonC.
6) Which lines have a y-intercept at left parenthesis, 0, comma, 4, right parenthesis?

Remember that in the general form y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE, the y-intercept is given by start color greenE, b, end color greenE.
• y, equals, start color maroonC, minus, 3, end color maroonC, x, start color greenE, plus, 4, end color greenE has a y-intercept at left parenthesis, 0, comma, start color greenE, 4, end color greenE, right parenthesis, so this is a correct answer.
• y, equals, start color maroonC, 4, end color maroonC, x, start color greenE, plus, 7, end color greenE has a y-intercept at left parenthesis, 0, comma, start color greenE, 7, end color greenE, right parenthesis.
• y, equals, start color greenE, 5, end color greenE, start color maroonC, plus, 4, end color maroonC, x has a y-intercept at left parenthesis, 0, comma, start color greenE, 5, end color greenE, right parenthesis.
• y, equals, start color greenE, 4, end color greenE, start color maroonC, minus, end color maroonC, x has a y-intercept at left parenthesis, 0, comma, start color greenE, 4, end color greenE, right parenthesis, so this is a correct answer.

### Refleksion spørgsmål

7) How do we find the slope of a line that is given in slope-intercept form?

The slope is always the coefficient of x, regardless of whether the x-term comes before or after the constant term.
For example, in y, equals, start color maroonC, 2, end color maroonC, x, start color greenE, plus, 1, end color greenE, the slope is start color maroonC, 2, end color maroonC, and in y, equals, start color greenE, 10, end color greenE, start color maroonC, minus, 100, end color maroonC, x, the slope is start color maroonC, minus, 100, end color maroonC.

### Udfordrende opgaver

8*) Which of these can be the equation of the line?

We don't have exact values on the graph, but we can see that y decreases as x increases, which means the line has a negative slope.
Furthermore, we can see that the line intersects the y-axis above the x-axis, which means the y-intercept is positive.
Out of the given equations, only y, equals, start color maroonC, minus, 2, end color maroonC, x, start color greenE, plus, 3, end color greenE has a negative slope left parenthesis, start color maroonC, minus, 2, end color maroonC, right parenthesis and a positive y-intercept left parenthesis, start color greenE, 3, end color greenE, right parenthesis.
Konklusion, det korrekte svar er y, equals, minus, 2, x, plus, 3.
9*) Write an equation of a line whose slope is 10 and y-intercept is left parenthesis, 0, comma, minus, 20, right parenthesis.

Remember that in the general form y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE, the slope is given by start color maroonC, m, end color maroonC, and the y-intercept is given by start color greenE, b, end color greenE.
Since we are given that the slope is start color maroonC, 10, end color maroonC and the y-intercept is left parenthesis, 0, comma, start color greenE, minus, 20, end color greenE, right parenthesis, the equation of the line is y, equals, start color maroonC, 10, end color maroonC, x, start color greenE, minus, 20, end color greenE.

# Hvorfor virker dette?

You might be wondering how it is that in slope-intercept form, start color maroonC, m, end color maroonC gives the slope and start color greenE, b, end color greenE gives the y-intercept.
Can this be some sort of magic? Well, it certainly is not magic. In math, there's always a justification. In this section we'll take a look at this property using the equation y, equals, start color maroonC, 2, end color maroonC, x, plus, start color greenE, 1, end color greenE as an example.

### Why $\greenE{b}$start color greenE, b, end color greenE gives the $y$y-intercept

At the y-intercept, the x-value is always zero. So if we want to find the y-intercept of y, equals, start color maroonC, 2, end color maroonC, x, plus, start color greenE, 1, end color greenE, we should substitute x, equals, 0 and solve for y.
\begin{aligned}y&=\maroonC{2}x+\greenE{1}\\\\ &=\maroonC{2}\cdot 0+\greenE{1}&\gray{\text{Indsæt }x=0}\\\\ &=0+\greenE{1}\\\\ &=\greenE{1}\end{aligned}
We see that at the y-intercept, start color maroonC, 2, end color maroonC, x becomes zero, and therefore we are left with y, equals, start color greenE, 1, end color greenE.
If we want to find the y-intercept of y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE, we should substitute x, equals, 0 and solve for y.
\begin{aligned}y&=\maroonC{m}x+\greenE{b}\\\\ &=\maroonC{m}\cdot 0+\greenE{b}&\text{Substitute }x=0.\\\\ &=0+\greenE{b}\\\\ &=\greenE{b}\end{aligned}
We see that at the y-intercept, start color maroonC, m, end color maroonC, x becomes zero, and we are therefore left with y, equals, start color greenE, b, end color greenE. For this reason, start color greenE, b, end color greenE will always give the y-intercept.

### Hvorfor $\maroonC{m}$start color maroonC, m, end color maroonC angiver hældningen

Let's refresh our memories about what slope is exactly. Slope is the ratio of the change in y over the change in x between any two points on the line.
$\text{Hældning}=\dfrac{\text{Ændring i }y}{\text{Ændring i }x}$
Hvis vi tager to punkter, hvor ændringen i x er præcis 1 enhed, vil så ændringen i y være lig med hældningen selv.
$\text{Hældning}=\dfrac{\text{Ændring i }y}{1}=\text{Ændring i }y$
Now let's look at what happens to the y-values in the equation y, equals, start color maroonC, 2, end color maroonC, x, plus, start color greenE, 1, end color greenE as the x-values constantly increase by 1 unit.
xspacey
0start color greenE, 1, end color greenE, plus, 0, dot, start color maroonC, 2, end color maroonCequals, start color greenE, 1, end color greenE
1start color greenE, 1, end color greenE, plus, 1, dot, start color maroonC, 2, end color maroonCequals, start color greenE, 1, end color greenE, plus, start color maroonC, 2, end color maroonC
2start color greenE, 1, end color greenE, plus, 2, dot, start color maroonC, 2, end color maroonCequals, start color greenE, 1, end color greenE, plus, start color maroonC, 2, end color maroonC, plus, start color maroonC, 2, end color maroonC
3start color greenE, 1, end color greenE, plus, 3, dot, start color maroonC, 2, end color maroonCequals, start color greenE, 1, end color greenE, plus, start color maroonC, 2, end color maroonC, plus, start color maroonC, 2, end color maroonC, plus, start color maroonC, 2, end color maroonC
4start color greenE, 1, end color greenE, plus, 4, dot, start color maroonC, 2, end color maroonCequals, start color greenE, 1, end color greenE, plus, start color maroonC, 2, end color maroonC, plus, start color maroonC, 2, end color maroonC, plus, start color maroonC, 2, end color maroonC, plus, start color maroonC, 2, end color maroonC
We see that each time x increases by 1 unit, y increases by start color maroonC, 2, end color maroonC units. This is because x determines the multiple of start color maroonC, 2, end color maroonC in the calculation of y.
As stated above, the change in y that corresponds to x increasing by 1 unit is equal to the slope of the line. For this reason, the slope is start color maroonC, 2, end color maroonC.
Let's look at what happens to the y-values in the general equation y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE as the x-values constantly increase by 1 unit.
xspacey
0start color greenE, b, end color greenE, plus, 0, dot, start color maroonC, m, end color maroonCequals, start color greenE, b, end color greenE
1start color greenE, b, end color greenE, plus, 1, dot, start color maroonC, m, end color maroonCequals, start color greenE, b, end color greenE, plus, start color maroonC, m, end color maroonC
2start color greenE, b, end color greenE, plus, 2, dot, start color maroonC, m, end color maroonCequals, start color greenE, b, end color greenE, plus, start color maroonC, m, end color maroonC, plus, start color maroonC, m, end color maroonC
3start color greenE, b, end color greenE, plus, 3, dot, start color maroonC, m, end color maroonCequals, start color greenE, b, end color greenE, plus, start color maroonC, m, end color maroonC, plus, start color maroonC, m, end color maroonC, plus, start color maroonC, m, end color maroonC
4start color greenE, b, end color greenE, plus, 4, dot, start color maroonC, m, end color maroonCequals, start color greenE, b, end color greenE, plus, start color maroonC, m, end color maroonC, plus, start color maroonC, m, end color maroonC, plus, start color maroonC, m, end color maroonC, plus, start color maroonC, m, end color maroonC
Vi kan se, at hver gang x stiger med 1 enhed, så stiger y med start color maroonC, m, end color maroonC enheder. Dette er fordi x bestemmer multiplum af start color maroonC, m, end color maroonC i beregningen af y.
Som anført ovenfor, svarer ændringen i y når x stiger med 1 enhed til linjens hældning. Af denne grund angiver start color maroonC, m, end color maroonC altid hældningen.

# Udfordrende opgave

Skriv ligningen for linjen.

We can write the line in slope-intercept form y, equals, start color maroonC, m, end color maroonC, x, plus, start color greenE, b, end color greenE. To do that, we need to find the y-intercept and the slope.
• The line intersects the y-axis at left parenthesis, 0, comma, start color greenE, minus, 3, end color greenE, right parenthesis.
• Efter left parenthesis, 0, comma, minus, 3, right parenthesis går linjen gennem left parenthesis, 1, comma, 1, right parenthesis. Når x stiger med 1 enhed, stiger y med 4 enheder. Det betyder, at linjens hældning er start color maroonC, 4, end color maroonC.
From the above, we find that start color greenE, b, end color greenE, equals, start color greenE, minus, 3, end color greenE and start color maroonC, m, end color maroonC, equals, start color maroonC, 4, end color maroonC.
Therefore, the equation of the line is y, equals, start color maroonC, 4, end color maroonC, x, start color greenE, minus, 3, end color greenE.