Review the graphs of a system of two linear equations in two variables: x - 5y = 7 and 3x - 3y + 4= Find the surullion to both equations
To analyze the graphs of the system of two linear equations, we first need to put them in slope-intercept form, which is y = mx + b.
For the first equation, x - 5y = 7, we can isolate y by subtracting x from both sides:
-5y = -x + 7.
Next, we divide both sides by -5:
y = (1/5)x - 7/5.
So, the slope of the first equation is 1/5 and the y-intercept is -7/5.
For the second equation, 3x - 3y + 4 = 0, we isolate y by subtracting 3x and 4 from both sides:
-3y = -3x - 4.
Next, we divide both sides by -3:
y = x + 4/3.
So, the slope of the second equation is 1 and the y-intercept is 4/3.
Now we can graph these equations:
The graph of y = (1/5)x - 7/5 represents a line with a positive slope of 1/5. It intersects the y-axis at -7/5, which is a little below y = -1. The line is increasing as x increases, with a steepness of 1 unit vertically for every 5 units horizontally.
The graph of y = x + 4/3 represents a line with a slope of 1. It intersects the y-axis at 4/3, which is slightly above y = 1. The line is increasing as x increases, with a steepness of 1 unit vertically for every 1 unit horizontally.
To find the solution to both equations, we need to find the point where the two graphs intersect. From the graphs, it is clear that the lines intersect at the point (1, 9/3) or (1, 3).
Therefore, the solution to the system of equations is x = 1 and y = 3.
To find the solution to a system of linear equations, we need to find the values of x and y that satisfy both equations simultaneously. Let's review the graphs of the given system of equations:
Equation 1: x - 5y = 7
Equation 2: 3x - 3y + 4 = 0
To graph Equation 1, we can rearrange it in slope-intercept form (y = mx + b) by isolating y:
x - 5y = 7
-5y = -x + 7
y = (1/5)x - 7/5
The graph of Equation 1 is a line with a slope of 1/5 and a y-intercept of -7/5.
To graph Equation 2, we can rearrange it in slope-intercept form:
3x - 3y + 4 = 0
-3y = -3x - 4
y = x + 4/3
The graph of Equation 2 is also a line with a slope of 1 and a y-intercept of 4/3.
Now let's consider the two graphs together:
The graph of Equation 1 (y = (1/5)x - 7/5) is a line that goes through the y-axis at -7/5 and has a positive slope. It slopes upward from left to right.
The graph of Equation 2 (y = x + 4/3) is a line with a y-intercept of 4/3 and a positive slope. It also slopes upward from left to right.
Based on the graphs, we can see that the lines intersect at a single point, indicating a unique solution for this system of equations. To find the exact coordinates of the point of intersection, we can set the two equations equal to each other:
(1/5)x - 7/5 = x + 4/3
Now, we can solve for x:
(1/5)x - x = 4/3 + 7/5
(1/5 - 1)x = 20/15 + 21/15
(-4/5)x = 41/15
Multiplying both sides by -5/4:
x = (41/15) * (-5/4)
x = -41/12
Substituting the value of x back into Equation 1, we can solve for y:
y = (1/5)(-41/12) - 7/5
y = -41/60 - 84/60
y = -125/60
y = -5/12
Therefore, the solution to the system of equations is x = -41/12 and y = -5/12.
To find the solution to the system of equations, we first need to analyze the graphs of the two equations.
The first equation is x - 5y = 7, which can be rewritten in slope-intercept form as y = (1/5)x - 7/5. This equation represents a line with a slope of 1/5 and a y-intercept of -7/5 (or a point where the line intersects the y-axis).
The second equation is 3x - 3y + 4 = 0, which can be rewritten in slope-intercept form as y = (1/3)x + 4/3. This equation represents a line with a slope of 1/3 and a y-intercept of 4/3.
Now, we can analyze the graphs of these two lines:
- The line represented by the equation y = (1/5)x - 7/5 has a positive slope, which means it slopes upward from left to right. It intersects the y-axis at -7/5, which is a point below the origin. So, the line passes through the y-axis below the origin and goes upward at an angle.
- The line represented by the equation y = (1/3)x + 4/3 also has a positive slope, which means it slopes upward from left to right. It intersects the y-axis at 4/3, which is a point above the origin. So, the line passes through the y-axis above the origin and goes upward at an angle.
To find the solution to this system of equations, we need to find the point where these two lines intersect. This point will be the shared solution for both equations.
The solution is the point of intersection of the lines, which is where the x-coordinate of one line is equal to the x-coordinate of the other line, and the y-coordinate of one line is equal to the y-coordinate of the other line.
To find the exact coordinates of the point of intersection, we can set the two equations equal to each other and solve for x. Let's do that:
(1/5)x - 7/5 = (1/3)x + 4/3
Now, we can solve for x:
(1/5)x - (1/3)x = 4/3 + 7/5
(3/15)x - (5/15)x = (20/15) + (21/15)
(-2/15)x = 41/15
To solve for x, we multiply both sides of the equation by -15/2:
x = (41/15) * (-15/2)
x = -41/2
Now, we substitute this value of x back into one of the equations, let's use the first equation x - 5y = 7:
(-41/2) - 5y = 7
Simplifying further:
-41 - 10y = 14
-10y = 55
y = -55/10
y = -11/2
So, the solution to the system of equations is x = -41/2 and y = -11/2, or (-41/2, -11/2) in coordinate form.
Therefore, the intersection point of the two lines represents the solution to the system of equations.