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A system of equations that form lines that intersect at one point.
Intersecting lines coincide at one point.
They have one solution.
They do not have the same slope or y-intercept.
GRAPHING INTERECEPTING LINES
Explanation of how a system of equations form intersecting lines
Slope-intercept
y = mx + b
y = -x + 6
y = x + 2
FIRST STEP: CREATE INITIAL LINEAR LINES
X- & Y-INTERCEPT TECHNIQUE
Discover the x-intercept and the y-intercept
Equation #1
X & Y- Intercept
y = mx + b
y = -x + 6
Discover the y-intercept
y = -x + 6
if x = 0, then
y = 0 + 6
y = 6
y-intercept = (0, 6)
Discover the x-intercept
y = -x + 6
if y = 0, then
0 = -x + 6
-6 + 0 = -x + 6 - 6
-6 = -x
6 = x
x-intercept = (6, 0)
-1 -1
Equation #2
X & Y- Intercept
y = mx + b
y = x + 2
Discover the y-intercept
y = x + 2
if x = 0, then
y = 0 + 2
y = 2
y-intercept = (0, 2)
Discover the x-intercept
y = x + 2
if y = 0, then
0 = x + 2
0 - 2 = x + 2 - 2
-2 = x
x-intercept = (-2, 0)
Place the x-intercept and y-intercept points on the graph to create linear lines
4
Equation #1:
4
2
(0, b) = (0, 6)
-4
(a, 0) = (6, 0)
-2
-2
2
6
-6
6
Equation #2:
(0, b) = (0, 2)
4
-4
-6
4
2
(a, 0) = (-2, 0)
-2
-2
2
6
SECOND STEP: DISCOVER THE SLOPE
SLOPE-INTERCEPT TECHNIQUE
Discover the slope from the slope-interecept equation
Equation #1
Slope-intercept
y = mx + b
y = -x + 6
Discover the slope
Since,
y-intercept = b, then,
y = b = 6 = y-intercept (0, 6)
therefore, coefficient of x = 1,
slope = m = 1
Equation #2
Slope-intercept
y = mx + b
y = x + 2
Discover the slope
Since,
y-intercept = b,
then,
y = b = 2 = y-intercept (0, 2)
therefore,
coefficient of x = 1,
slope = m = 1
Draw the slope for each graph. This assist in ensuring the right linear direction.
-6
4
(6, 0)
6
2
-4
-2
-2
2
Equation #1:
4
(0, 6)
1 unit
1 unit
(1, 5)
(2, 4)
Equation #2:
6
4
-4
-6
(-2, 0)
-2
-2
2
4
2
1 unit
1 unit
(1, 3)
(2, 4)
1 unit
(0, 2)
SOLUTION
The slope and x- & y intercept are a good way to quickly observe system of equations.
Rule:
Both slopes are not equal
Both y-intercepts are not equal
-6
4
6
4
2
(2, 4)
-4
-2
-2
2
m = -1
Equation #1
y = -x + 6
m = 1
Equation #1
y = x + 2
Equation #1
slope = -1
y-intercept = (0,6)
Equation #2
slope = 3
y-intercept = (0, 2)
Solution: (2, 4)
intersecting point
Equation #1
slope = 1
y-intercept = (0, 0)
Equation #2
slope = -1
y-intercept = (0, 5)
Intersecting equations have lines that intersect at one point on a graph.
The slope and y-intercepts' are not the same.
The two linear lines form independent equations.
Although independent, the y-value depends on the x-value to solve the equation at one point.
Therefore, the equations are consistent at one point.
Graphically these linear lines INTERSECT
at
one point
thus, there is
one solution
You are observing
intersecting lines.
m
(x - )
x
1
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