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 linear equations, let's first rearrange each equation to solve for y:
Equation 1: x - 5y = 7
Solving for y:
-5y = -x + 7
y = (1/5)x - (7/5)
Equation 2: 3x - 2y = -4
Solving for y:
-2y = -3x - 4
y = (3/2)x + 2
Now we have both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
From the slope-intercept equations, we can see that the slope of the first equation is 1/5, and the slope of the second equation is 3/2. This tells us that the lines are not parallel.
To find the solution to the system of equations, we can set the two equations equal to each other and solve for x:
(1/5)x - (7/5) = (3/2)x + 2
To get rid of the fractions, we can multiply both sides of the equation by 10 to clear the denominators:
2x - 14 = 15x + 20
Now, we can solve for x:
-13x = 34
Dividing both sides by -13 gives:
x = -34/13
To find the corresponding y value, we can substitute this value of x into either of the original equations. Let's use the first equation:
x - 5y = 7
(-34/13) - 5y = 7
To solve for y, we can multiply both sides of the equation by -13/5:
-2(34/13) + (13/5)(-5y) = (7)(-13/5)
-68/13 - 13y = -91/5
To combine like terms, we need to get a common denominator:
-340/65 - 13y = -254/25
Multiplying everything by 25 to clear the denominators, we have:
-3400/65 - 325y = -254
-3400 - 325y = -254 * 65
-3400 - 325y = -16510
To solve for y, we can subtract -3400 from both sides:
-325y = -16510 + 3400
-325y = -13110
Dividing both sides by -325 gives:
y = 13110/325
Simplifying the fraction, we find:
y = 42
Therefore, the solution to the system of equations is x = -34/13 and y = 42.
To graph the system of equations, we plot the points (-34/13, 42) as the intersection point of the two lines:
(Note: The accuracy of the graph may vary depending on the scale and precision.)
The graph of the system of equations shows that the lines intersect at the point (-34/13, 42), which confirms our solution.
To review the graphs of a system of two linear equations in two variables, let's first rearrange the equations to the slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
Given equations:
1) x - 5y = 7
2) 3x - 2y = -4
Rearranging equation 1:
x - 5y = 7
-5y = -x + 7
y = (1/5)x - (7/5)
Rearranging equation 2:
3x - 2y = -4
-2y = -3x - 4
y = (3/2)x + 2
Now, we can graph the equations:
For equation 1, with slope (1/5) and y-intercept (-7/5), we can plot two points to draw a line. Let's choose x = 0 and x = 5, for convenience.
For x = 0: y = (1/5)(0) - (7/5) = -7/5 --> point (0, -7/5)
For x = 5: y = (1/5)(5) - (7/5) = -6/5 --> point (5, -6/5)
For equation 2, with slope (3/2) and y-intercept 2, again we can plot two points.
For x = 0: y = (3/2)(0) + 2 = 2 --> point (0, 2)
For x = 5: y = (3/2)(5) + 2 = 9.5 --> point (5, 9.5)
Now we can plot these points on a graph:
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The first equation is represented by a line passing through the points (0, -7/5) and (5, -6/5), and the second equation is represented by a line passing through the points (0, 2) and (5, 9.5).
To find the solution to both equations, we need to find the point where the two lines intersect. We can do this by setting the two equations equal to each other and solving for x:
(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
Rearranging and simplifying:
15x - 2x = -20 + 14
13x = -6
x = -6/13
Now, substitute this value of x back into either equation to find the corresponding value of y.
Using equation 1:
y = (1/5)(-6/13) - (7/5)
y = -6/65 - 91/65
y = -97/65
Therefore, the solution to the system of equations is (x, y) = (-6/13, -97/65).
To review the graphs of a system of two linear equations, we need to graph each equation and see where they intersect. The point of intersection will be the solution to both equations. Let's solve the system of equations step by step:
Equation 1: x - 5y = 7
To graph this equation, we need to rearrange it into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
x - 5y = 7
-5y = -x + 7
y = (1/5)x - 7/5
Equation 2: 3x - 2y = -4
To graph this equation, we also need to rearrange it into slope-intercept form.
3x - 2y = -4
-2y = -3x - 4
y = (3/2)x + 2
Now that we have both equations in slope-intercept form, we can graph them on the same coordinate plane.
To graph the first equation (y = (1/5)x - 7/5), start by plotting the y-intercept at (0, -7/5). Then, use the slope (1/5) to find another point by moving 5 units to the right and 1 unit up. Connect the two points with a straight line.
To graph the second equation (y = (3/2)x + 2), start by plotting the y-intercept at (0, 2). Then, use the slope (3/2) to find another point by moving 2 units to the right and 3 units up. Connect the two points with a straight line.
The two lines will intersect at a point. That point represents the solution to both equations. Locate the point of intersection, and the coordinates of that point will be the solution to the system of equations.