Answer :
Answer:
To show graphically that the system of equations
3
+
2
=
7
3x+2y=7 and
6
+
4
=
5
6x+4y=5 has no solutions, let's analyze and graph these equations step by step.
Step 1: Graphing
3
+
2
=
7
3x+2y=7
First, rewrite
3
+
2
=
7
3x+2y=7 in slope-intercept form (
=
+
y=mx+b) to find its graph.
2
=
−
3
+
7
2y=−3x+7
=
−
3
2
+
7
2
y=−
2
3
x+
2
7
Now, graph this line:
Find intercepts:
Y-intercept: When
=
0
x=0,
=
7
2
=
3.5
y=
2
7
=3.5. So, one point is
(
0
,
3.5
)
(0,3.5).
X-intercept: When
=
0
y=0,
−
3
2
+
7
2
=
0
−
2
3
x+
2
7
=0.
−
3
2
=
−
7
2
−
2
3
x=−
2
7
=
7
3
≈
2.33
x=
3
7
≈2.33
So, another point is approximately
(
7
3
,
0
)
(
3
7
,0).
Plot the points and draw the line:
Connect the points
(
0
,
3.5
)
(0,3.5) and
(
7
3
,
0
)
(
3
7
,0) to graph the line.
Step 2: Graphing
6
+
4
=
5
6x+4y=5
Similarly, rewrite
6
+
4
=
5
6x+4y=5 in slope-intercept form.
4
=
−
6
+
5
4y=−6x+5
=
−
3
2
+
5
4
y=−
2
3
x+
4
5
Now, graph this line:
Find intercepts:
Y-intercept: When
=
0
x=0,
=
5
4
=
1.25
y=
4
5
=1.25. So, one point is
(
0
,
1.25
)
(0,1.25).
X-intercept: When
=
0
y=0,
−
3
2
+
5
4
=
0
−
2
3
x+
4
5
=0.
−
3
2
=
−
5
4
−
2
3
x=−
4
5
=
5
3
≈
1.67
x=
3
5
≈1.67
So, another point is approximately
(
5
3
,
0
)
(
3
5
,0).
Plot the points and draw the line:
Connect the points
(
0
,
1.25
)
(0,1.25) and
(
5
3
,
0
)
(
3
5
,0) to graph the line.
Step 3: Analyzing the Graphs
Now, look at the graphs of both lines on the same coordinate system.
The first line
3
+
2
=
7
3x+2y=7 has a slope of
−
3
2
−
2
3
and intercepts as plotted.
The second line
6
+
4
=
5
6x+4y=5 also has the same slope
−
3
2
−
2
3
and intercepts as plotted.
Since the equations of both lines are multiples of each other (the second equation is exactly twice the first equation), they represent the same line. Therefore, they are not distinct lines but essentially the same line.
Conclusion
Since both equations represent the same line, they intersect at infinitely many points (all points on the line itself), rather than having no intersection points. Therefore, the system
3
+
2
=
7
3x+2y=7 and
6
+
4
=
5
6x+4y=5 does not have no solutions; it has infinitely many solutions.
Coordinates of the Graph
Both equations
3
+
2
=
7
3x+2y=7 and
6
+
4
=
5
6x+4y=5 graph to the same line with coordinates:
(
0
,
3.5
)
(0,3.5) (Y-intercept of
3
+
2
=
7
3x+2y=7)
(
7
3
,
0
)
(
3
7
,0) (Approximate X-intercept of
3
+
2
=
7
3x+2y=7)
(
0
,
1.25
)
(0,1.25) (Y-intercept of
6
+
4
=
5
6x+4y=5)
(
5
3
,
0
)
(
3
5
,0) (Approximate X-intercept of
6
+
4
=
5
6x+4y=5)
These points lie on the same line, confirming that the system has infinitely many solutions.