Review the graphs of a system of two linear equations in two variables: x−5y=7 and 3x−2y=−4 . Find the solution to both equations.
To review the graphs of the system of equations x−5y=7 and 3x−2y=−4, we can rewrite both equations in slope-intercept form (y = mx + b).
1. For the first equation, we have x − 5y = 7. To rewrite it in slope-intercept form, we isolate y:
-5y = -x + 7
y = (1/5)x - 7/5
2. For the second equation, we have 3x − 2y = -4. Isolating y, we get:
-2y = -3x - 4
y = (3/2)x + 2
Now, let's plot the graphs of both equations:
Graph 1:
The equation y = (1/5)x - 7/5 represents a line with a slope of 1/5 and a y-intercept of -7/5. When x = 0, y = -7/5, and when y = 0, x = 7. Therefore, we have the points (0, -7/5) and (7, 0). Drawing a line passing through these points, we get:
|
|
| * *
|
---------------------------
|
|
Graph 2:
For the equation y = (3/2)x + 2, the slope is 3/2, and the y-intercept is 2. When x = 0, y = 2, and when y = 0, x = -4/3. So, we have the points (0, 2) and (-4/3, 0). Drawing the line passing through these points, we get:
*
|
|
-----------------
|
|
The solution to the system of equations can be found by identifying the point of intersection on the graph. Looking at the graphs, we can see that the lines intersect at the point (-3, -1).
Therefore, the solution to both equations is x = -3 and y = -1.
To review the graphs of the system of two linear equations x−5y=7 and 3x−2y=−4, we will first need to convert both equations into slope-intercept form (y = mx + b).
For the first equation, x−5y=7, we need to isolate y:
x − 5y = 7
-5y = -x + 7
y = (1/5)x - (7/5)
For the second equation, 3x−2y=−4, we isolate y as well:
3x − 2y = -4
-2y = -3x - 4
y = (3/2)x + 2
Now that we have the equations in slope-intercept form, we can plot the graphs on a Cartesian coordinate system.
The graph of the first equation, y = (1/5)x - (7/5), intersects the y-axis at -7/5 and has a slope of 1/5.
The graph of the second equation, y = (3/2)x + 2, intersects the y-axis at 2 and has a slope of 3/2.
To find the solution to both equations, we need to find the intersection point of these two lines.
Equating the two equations:
(1/5)x - (7/5) = (3/2)x + 2
Multiplying both sides of the equation by 10 to eliminate the fractions:
2x - 14 = 15x + 20
Subtracting 2x from both sides and subtracting 20 from both sides:
-14 - 20 = 15x - 2x
-34 = 13x
Dividing both sides by 13:
x = -34/13
Substituting this value of x back into one of the original equations, let's use x−5y=7:
(-34/13) − 5y = 7
Multiplying both sides by 13 to eliminate fractions:
-34 - 65y = 91
Subtracting -34 from both sides:
-65y = 125
Dividing both sides by -65:
y = -125/65
Simplifying, we get x = -34/13 and y = -25/13.
Therefore, the solution to both equations is (x, y) = (-34/13, -25/13).
To review the graphs of a system of two linear equations in two variables, we need to graph each equation and find the point where they intersect. This point represents the solution to both equations. Here's how we can do it:
1. Graph the first equation: x - 5y = 7
- First, convert the equation to slope-intercept form (y = mx + b) by isolating y:
x - 5y = 7
-5y = -x + 7
y = (1/5)x - 7/5
- Plot the y-intercept (0, -7/5)
- Use the slope (1/5) to find more points. For example, if you go right 5 units and up 1 unit from the y-intercept, you'll get another point (5, -6/5).
- Connect the points to draw the graph of the first equation.
2. Graph the second equation: 3x - 2y = -4
- Convert the equation to slope-intercept form (y = mx + b) by isolating y:
3x - 2y = -4
-2y = -3x - 4
y = (3/2)x + 2
- Plot the y-intercept (0, 2)
- Use the slope (3/2) to find more points. For example, if you go right 2 units and up 3 units from the y-intercept, you'll get another point (2, 5).
- Connect the points to draw the graph of the second equation.
3. Find the point of intersection:
- Look at the graphs of the two equations. The point where they intersect represents the solution to both equations.
- In this case, it appears that the two lines intersect at the point (3, -1).
Therefore, the solution to both equations is x = 3 and y = -1.