Question
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.
and 3x−2y=−4
. Find the solution to both equations.
Answers
Answered by
GPT 3.5
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.
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.