We want to
solve the given system of equations by graphing and locating the point of intersection (if it
exists). The system:
y = x - 1
y = 3x -
9
A system of two linear equations in two variables can
have 0, 1, or infinite solutions. If the lines are parallel, there are no solutions. (The system
is said to be inconsistent.) If the lines intersect at a single point, there is one solution.
(The system is said to be consistent and independent.) If the two lines are actually the same
line, there are an infinite number of solutions for the system. (The system is said to be
consistent and dependent.)
The first line, y = x - 1, has slope 1 and
y-intercept of -1. This means that the line includes the point (0, -1), as this is the
y-intercept. To find another point on the line we can use the slope: for every unit to the
right, we go we should go up 1 unit. Points on this line include (0,-1), (1,0), (2,1), (3,2),
and so on.
The second line, y = 3x - 9, has slope 3 and y-intercept -9. Once
we plot (0,-9) (the y-intercept), for every 1 unit to the right we must go up 3 units. Points on
this line include (0,-9), (1,-6), (2,-3),(3,0), and so on.
The graphs: y = x
- 1 in blue and y = 3x - 9 in red (see attached).
It
appears that the lines intersect at (4,3). We can verify this algebraically:
If x = 4 then y = x - 1 = 4 - 1 = 3
If x = 4 then y = 3x -
9 = 3(4) - 9 = 3 as required.
The solution
is the point (4,3).
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