• Answer:
[tex] \Large{\boxed{\sf y = 5x - 21 }} [/tex]
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• Explanation:
Let's assume that we want to find the equation of the line that passes through A(5 , 4) and B(8 , 19) in slope-intercept form.
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[tex] \Large{\left[ \begin{array}{c c c} \underline{\tt Slope-Intercept \: Form \text{:}} \\ ~ \\ \tt y = mx + b \end{array} \right] } [/tex]
Where:
• m is the slope of the line.
• b is its y-intercept.
• (x , y) is a point on the line.
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The first thing we have to do is calculate the slope of the line. To do that, we will use the slope formula, which is the following:
[tex] \left[ \begin{array}{c c c} \tt ~ \\ \tt m = \dfrac{\Delta y}{\Delta x} = \dfrac{y_B - y_{A}}{x_B - x_A}\\ \tt ~ \end{array} \right] [/tex]
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Let's identify our values and apply the formula:
[tex] \sf A(\underbrace{\sf 5}_{x_A} \ , \ \overbrace{\sf 4}^{y_A}) \ \ and \ \ B(\underbrace{\sf 8}_{x_B} \ , \ \overbrace{\sf 19}^{y_B}) \\ \\ \\ \sf \implies \sf m = \dfrac{19 - 4}{8 - 5} \\ \\ \implies \sf m = \dfrac{15}{3} = \dfrac{3 \ast 5}{3}\\ \\ \implies \boxed{\sf m = 5} [/tex]
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Substituting the value of the slope, the equation of the line becomes:
[tex] \sf y = 5c + b [/tex]
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Since the coordinates of all points on a line satisfy its equation, we can substitute the coordinates of one of the two given points into the equation and solve for b.
[tex] \sf A(\underbrace{\sf 5}_{x} \ , \ \overbrace{\sf 4}^{y} ) \\ \\ \implies \sf 4 = 5(5) + b \\ \\ \implies \sf 4 = 25 + b \Longleftrightarrow \boxed{\sf b = -21} [/tex]
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Therefore, the equation of the line in slope-intercept form is:[tex] \boxed{\boxed{\sf y = 5x - 21}} [/tex]
Answer:
Step-by-step explanation:
This question is asking us to determine an equation for a straight line, given that it passes through the points (5,4) and (8,19). The answer should be written in slope-intercept form (also known as y = mx + c or y = mx + b).
So first, we're going to apply our slope formula to determine the slope of this line.
The slope formula is:
[tex]\bigstar\quad\large\boldsymbol{m=\cfrac{y_2-y_1}{x_2-x_1}}[/tex]
The first point is (5,4).
The second point is (8,19).
Substitute the values:
[tex]\sf m=\cfrac{19-4}{8-5}[/tex]
[tex]\sf m=\cfrac{15}{3}[/tex]
[tex]\sf m=5[/tex]
Therefore, the slope of this line is 5.
[tex]\dotfill[/tex]
Now we'll determine the line's equation. Start by using the point-slope formula:
[tex]\large\bigstar\quad\boldsymbol{y-y_1=m(x-x_1)}[/tex]
You can use any point, but I'll choose the first one: (5,4)
Substitute these values:
[tex]\to\quad\sf y-4=5(x-5)[/tex]
[tex]\to\quad\sf y-4=5x-25[/tex]
[tex]\to\quad\sf y=5x-25+4[/tex]
[tex]\to\quad\sf y=5x-21[/tex]
Therefore, the equation of this line is: y = 5x - 21.
Have a nice day! :)