Graph two lines whose solution is 1,4
Answers
Answer:
Constructing a set of axes, we can first locate the two given points, (1,4) and (0,−1), to create our first line. Our second line can be any other line that passes through (1,4) but not (0,−1), so there are many possible answers. Below is one possible construction:
Two_lines_ebe47645683c432b697fd92eacb47ad9
Focusing first on the line through the two given points, we can find the slope of this line two ways:
Graphically, we can start at the point (0,−1) and then count how many units we go up divided by how many units we then go right to get to the point (1,4), as in the diagram below.
Two_lines_slope_1_4201287cb679a71d9daccc4a189c5229
This gives a slope of m=51=5.
Algebraically, we can find the difference between the y-coordinates of the two points, and divide it by the difference between the x-coordinates.
m=4−(−1)1−0=5.
Because the y-intercept of this line is -1, we have b=−1. So, the equation of our first line is y=5x−1.
We can reason in a similar way for our second line. Graphically, we see our second line contains the point (0,6), so we can start at the point (0,6) and then count how many units we go down divided by how many units we then go right to get to the point (1,4), as in the diagram below.
Two_lines_slope_2_738d1f8e2e397b56a95eef96655f4d97
This gives a slope of m=−21=−2.
We can also find the slope algebraically:
m=4−61−0=−2.
Because we have a y-intercept of 6, b=6. So, the equation of our first line is y=−2x+6.
Using this idea that a solution to a system of equations is a pair of values that makes both equations true, we decide that our system of equations does have a solution, because
The coordinates of every point on a line satisfy its equation, and
The point (1,4) lies on both lines.
We can confirm that (1,4) is our system's solution by substituting x=1 and y=4 into both equations:
4=5(1)−1
and
4=−2(1)+6.
Based on our work above, we can make a general observation that if a system of linear equations has a solution, that solution corresponds to the intersection point of the two lines because the coordinate pair naming every point on a graph is a solution to its corresponding equation.
Answer:
Step-by-step explanation:
A)To begin building a set of axes, we can use the two given points (1,4) and (0,1) to create our first line. Because our second line can be any other line that passes through (1,4) but not (0,1), there are numerous possible solutions. One possible structure is shown below:
B) We can find the slope of the line that connects the two given points in two ways:
To get to the point (1,4), we can start at (0,1) and count how many units we go up divided by how many units we go right, as shown in the diagram below.
This gives a slope of m=5/1=5.
Algebraically, we can find the difference between the y-coordinates of the two points, and divide it by the difference between the x-coordinates
m=
Because the y-intercept of this line is -1, we have b=−1. So, the equation of our first line is y=5x−1.
We can reason in a similar way for our second line. Graphically, we see our second line contains the point (0,6), so we can start at the point (0,6) and then count how many units we go down divided by how many units we then go right to get to the point (1,4), as in the diagram below.
slope of m=−21=−2.
Because we have a y-intercept of 6, b=6. So, the equation of our first line is y=−2x+6.
C) Using the concept that a solution to a system of equations is a pair of values that make both equations true, we determine that our system of equations has a solution, because
Every point on a line's coordinates satisfy its equation, and the point (1,4) lies on both lines.
By substituting x=1 and y=4 into both equations, we can confirm that (1,4) is our system's solution:
4 = 5 ( 1 ) - 1
&
4 = - 2 ( 1 ) +
Based on our previous work, we can conclude that if a system of linear equations has a solution, that solution corresponds to the intersection point of the two lines, because the coordinate pair naming each point on a graph is a solution to its corresponding equation.
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