Answer:
(a) y = 6x + 5
(b) y = 3x + 17
(c) Consistent Independent
(d) (4, 29)
Step-by-step explanation:
Part (a)
[tex]\sf let\:(x_1,y_1)=(2,17)[/tex]
[tex]\sf let\:(x_2,y_2)=(0,5)[/tex]
[tex]\sf slope\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{5-17}{0-2}=6[/tex]
[tex]\sf Slope\:intercept\:form\:of\:linear\:equation:y=mx+b[/tex]
(where m is the slope and b is the y-intercept)
y-intercept is when x = 0, therefore, y-intercept is 5
[tex]\sf \implies y=6x+5[/tex]
Part (b)
[tex]\sf let\:(x_1,y_1)=(1,20)[/tex]
[tex]\sf let\:(x_2,y_2)=(3,26)[/tex]
[tex]\sf slope\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{26-20}{3-1}=3[/tex]
[tex]\sf Point\:slope\:form\:of\:linear\:equation:y-y_1=m(x-x_1)[/tex]
[tex]\sf \implies y-20=3(x-1)[/tex]
[tex]\sf \implies y=3x+17[/tex]
Part (c)
- Consistent Independent: One solution (lines intersects at 1 point)
- Coincident: Infinite solutions (lines lie exactly on top of each other)
- Inconsistent: No solution (parallel lines)
The system of equations has to be Consistent Independent by process of elimination. As the two lines have different slopes, they are not parallel so there are not inconsistent. As the lines have different equations, they do not lie exactly on top of each other and so are not coincident.
Part (d)
Using the substitution method.
[tex]\sf Equation\:1:y=6x+5[/tex]
[tex]\sf Equation\:2: y=3x+17[/tex]
Substitute Equation 1 into Equation 2 and solve for x:
[tex]\sf \implies 6x+5=3x+17[/tex]
[tex]\sf \implies 3x=12[/tex]
[tex]\sf \implies x=4[/tex]
Substitute found value of x into Equation 1 and solve for y:
[tex]\sf \implies 6(4)+5=29[/tex]
Therefore, the solution to the system of equations is (4, 29) which is also the point of intersection of the two lines.
