To solve the system of equations, we can use the substitution or elimination method. Let's use the elimination method:
First, we can multiply the first equation by 3 and the second equation by -1 to create canceling terms for x:
3x - 15y = 21
-3x + 2y = 4
Adding the two equations together, we get:
-13y = 25
Dividing both sides by -13, we get:
y = -25/13
Substituting this value back into the first equation, we get:
x - 5(-25/13) = 7
x + 125/13 = 7
x = 7 - 125/13
x = (91 - 125)/13
x = -34/13
Therefore, the solution to both equations is (-34/13, -25/13).
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.
(1 point)
The intersection point is (
).
15 answers
are you sure
Apologies for the confusion. Let's recheck the solution.
We have the system of equations:
1) x - 5y = 7
2) 3x - 2y = -4
We can solve this system using the elimination method by multiplying equation 1 by 2 and equation 2 by 5 to create canceling terms for y:
2(x - 5y) = 2(7)
5(3x - 2y) = 5(-4)
Expanding these equations, we get:
2x - 10y = 14
15x - 10y = -20
Now, let's subtract equation 1 from equation 2:
(15x - 10y) - (2x - 10y) = -20 - 14
13x = -34
Dividing both sides by 13, we get:
x = -34/13
Substituting this value back into equation 1, we get:
(-34/13) - 5y = 7
Multiplying through by 13 to eliminate the fraction, we get:
-34 - 65y = 91
Simplifying, we have:
-65y = 125
Dividing both sides by -65, we get:
y = -125/65
Reducing the fraction, we have:
y = -25/13
Therefore, the solution to both equations is x = -34/13 and y = -25/13.
We have the system of equations:
1) x - 5y = 7
2) 3x - 2y = -4
We can solve this system using the elimination method by multiplying equation 1 by 2 and equation 2 by 5 to create canceling terms for y:
2(x - 5y) = 2(7)
5(3x - 2y) = 5(-4)
Expanding these equations, we get:
2x - 10y = 14
15x - 10y = -20
Now, let's subtract equation 1 from equation 2:
(15x - 10y) - (2x - 10y) = -20 - 14
13x = -34
Dividing both sides by 13, we get:
x = -34/13
Substituting this value back into equation 1, we get:
(-34/13) - 5y = 7
Multiplying through by 13 to eliminate the fraction, we get:
-34 - 65y = 91
Simplifying, we have:
-65y = 125
Dividing both sides by -65, we get:
y = -125/65
Reducing the fraction, we have:
y = -25/13
Therefore, the solution to both equations is x = -34/13 and y = -25/13.
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 solve the system of equations, we can use the substitution or elimination method. Let's use the substitution method:
From the first equation, we can solve for x:
x = 5y + 7
Substitute this expression for x into the second equation:
3(5y + 7) - 2y = -4
15y + 21 - 2y = -4
13y = -25
y = -25/13
Substitute this value for y back into the first equation to solve for x:
x - 5(-25/13) = 7
x + 125/13 = 7
x = 7 - 125/13
x = (91 - 125)/13
x = -34/13
Therefore, the solution to the system of equations is x = -34/13 and y = -25/13.
From the first equation, we can solve for x:
x = 5y + 7
Substitute this expression for x into the second equation:
3(5y + 7) - 2y = -4
15y + 21 - 2y = -4
13y = -25
y = -25/13
Substitute this value for y back into the first equation to solve for x:
x - 5(-25/13) = 7
x + 125/13 = 7
x = 7 - 125/13
x = (91 - 125)/13
x = -34/13
Therefore, the solution to the system of equations is x = -34/13 and y = -25/13.
What is the last step to solving a system of equations?(1 point)
Responses
Make a table of solutions of the linear equations.
Make a table of solutions of the linear equations.
Estimate the intersection point.
Estimate the intersection point.
Graph the lines.
Graph the lines.
Check the answer.
Responses
Make a table of solutions of the linear equations.
Make a table of solutions of the linear equations.
Estimate the intersection point.
Estimate the intersection point.
Graph the lines.
Graph the lines.
Check the answer.
Check the answer.
Solving the system y=−x+3 and x+y=8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain.
(1 point)
Responses
The statement is false, because the lines are parallel.
The statement is false, because the lines are parallel.
The statement is true, because the lines are parallel.
The statement is true, because the lines are parallel.
The statement is false, because the lines have an intersection point.
The statement is false, because the lines have an intersection point.
The statement is true, because the lines have an intersection point.
(1 point)
Responses
The statement is false, because the lines are parallel.
The statement is false, because the lines are parallel.
The statement is true, because the lines are parallel.
The statement is true, because the lines are parallel.
The statement is false, because the lines have an intersection point.
The statement is false, because the lines have an intersection point.
The statement is true, because the lines have an intersection point.
The statement is true, because the lines have an intersection point.
By graphing the two equations, we can see that the lines do intersect at a single point. The equation y = -x + 3 represents a downward diagonal line that intersects the y-axis at 3. The equation x + y = 8 represents a diagonal line with a positive slope that intersects the y-axis at 8. These lines intersect at the point (2, 6), where the x-coordinate is 2 and the y-coordinate is 6. Therefore, there is a solution to the system of equations.
By graphing the two equations, we can see that the lines do intersect at a single point. The equation y = -x + 3 represents a downward diagonal line that intersects the y-axis at 3. The equation x + y = 8 represents a diagonal line with a positive slope that intersects the y-axis at 8. These lines intersect at the point (2, 6), where the x-coordinate is 2 and the y-coordinate is 6. Therefore, there is a solution to the system of equations.
Does the graph show the system of equations x+y=2 and −x+y=7? Should the lines for the system be parallel?
(1 point)
Responses
Both graphs are correct. The lines should be parallel.
Both graphs are correct. The lines should be parallel.
The graph of −x+y=7 is incorrect. The lines should intersect.
The graph of negative x plus y equals 7 is incorrect. The lines should intersect.
The graph of −x+y=7 is incorrect. The lines should be parallel.
The graph of negative x plus y equals 7 is incorrect. The lines should be parallel.
The graph of x+y=2 is incorrect. The lines should intersect.
(1 point)
Responses
Both graphs are correct. The lines should be parallel.
Both graphs are correct. The lines should be parallel.
The graph of −x+y=7 is incorrect. The lines should intersect.
The graph of negative x plus y equals 7 is incorrect. The lines should intersect.
The graph of −x+y=7 is incorrect. The lines should be parallel.
The graph of negative x plus y equals 7 is incorrect. The lines should be parallel.
The graph of x+y=2 is incorrect. The lines should intersect.
The graph does not show the system of equations x + y = 2 and -x + y = 7 correctly.
The correct graph for x + y = 2 represents a line with a negative slope that intersects the y-axis at 2.
The correct graph for -x + y = 7 represents a line with a positive slope that intersects the y-axis at 7.
These lines are not parallel as they have different slopes. Instead, they intersect at the point (4, -2).
Therefore, the correct statement is:
The graph of -x + y = 7 is incorrect. The lines should intersect.
The correct graph for x + y = 2 represents a line with a negative slope that intersects the y-axis at 2.
The correct graph for -x + y = 7 represents a line with a positive slope that intersects the y-axis at 7.
These lines are not parallel as they have different slopes. Instead, they intersect at the point (4, -2).
Therefore, the correct statement is:
The graph of -x + y = 7 is incorrect. The lines should intersect.
Solve the given system of linear equations by inspection.
y=35x−5
y=−35x−5
(1 point)
Responses
infinitely many solutions
infinitely many solutions
(−5,0)
left parenthesis negative 5 comma 0 right parenthesis
no solution
no solution
(0,−5)
y=35x−5
y=−35x−5
(1 point)
Responses
infinitely many solutions
infinitely many solutions
(−5,0)
left parenthesis negative 5 comma 0 right parenthesis
no solution
no solution
(0,−5)
The given system of linear equations y = 35x - 5 and y = -35x - 5 has no solution.
This can be determined by inspecting the equations. The slopes of the two lines are 35 and -35, which means they are parallel and will never intersect.
Therefore, the correct response is:
no solution.
This can be determined by inspecting the equations. The slopes of the two lines are 35 and -35, which means they are parallel and will never intersect.
Therefore, the correct response is:
no solution.
are you sure
Apologies for the confusion in my previous response. Let's re-evaluate the system of equations.
The given system of linear equations is:
y = 35x - 5
y = -35x - 5
By comparing the two equations, we can see that both equations are equal to y. Therefore, the system of equations is actually representing the same line.
Since the lines are the same, there are infinitely many solutions. Any point on the line will satisfy both equations.
Therefore, the correct response is:
infinitely many solutions.
The given system of linear equations is:
y = 35x - 5
y = -35x - 5
By comparing the two equations, we can see that both equations are equal to y. Therefore, the system of equations is actually representing the same line.
Since the lines are the same, there are infinitely many solutions. Any point on the line will satisfy both equations.
Therefore, the correct response is:
infinitely many solutions.
12 answers