Question
Determine the number of solutions of this system of linear equations:
-x+y=3
-2x+2y=3
(Show how you got this answer)
-x+y=3
-2x+2y=3
(Show how you got this answer)
Answers
Answer
Can you explain the steps of how you fit this answer?
Answer
How would I write this equation? Could you simplify it a bit?
Answered by
GPT 3.5
We can solve this system of equations by rearranging the first equation to solve for x:
x = y - 3
Then substitute x into the second equation:
-2(y - 3) + 2y = 3
-2y + 6 + 2y = 3
6 = 3
Since this equation is false, there are no solutions to this system of linear equations. The two equations are parallel and will never intersect, meaning there is no point where they are both satisfied simultaneously.
x = y - 3
Then substitute x into the second equation:
-2(y - 3) + 2y = 3
-2y + 6 + 2y = 3
6 = 3
Since this equation is false, there are no solutions to this system of linear equations. The two equations are parallel and will never intersect, meaning there is no point where they are both satisfied simultaneously.
Answered by
GPT 3.5
Sure! To determine the number of solutions of a system of linear equations, we typically look at the number of solutions when the equations are graphed on a coordinate plane. In this case, since the equations are in a standard form (y = mx + b), we can easily rearrange them to graph the lines.
The first equation is -x + y = 3. Rearranging the equation to y = x + 3, we see that the y-intercept is 3 and the slope is 1 (since the coefficient of x is 1). So, the first line has a y-intercept of 3 and goes up by 1 unit for every 1 unit move to the right.
The second equation is -2x + 2y = 3. Rearranging the equation to y = x + 1.5, we see that the y-intercept is 1.5 and the slope is 1 (since the coefficient of x is 1). So, the second line also has a y-intercept of 1.5 and goes up by 1 unit for every 1 unit move to the right.
When we graph both of these lines, we can see that they are parallel, meaning they will never intersect. Since the lines do not intersect, there are no solutions to this system of linear equations.
The first equation is -x + y = 3. Rearranging the equation to y = x + 3, we see that the y-intercept is 3 and the slope is 1 (since the coefficient of x is 1). So, the first line has a y-intercept of 3 and goes up by 1 unit for every 1 unit move to the right.
The second equation is -2x + 2y = 3. Rearranging the equation to y = x + 1.5, we see that the y-intercept is 1.5 and the slope is 1 (since the coefficient of x is 1). So, the second line also has a y-intercept of 1.5 and goes up by 1 unit for every 1 unit move to the right.
When we graph both of these lines, we can see that they are parallel, meaning they will never intersect. Since the lines do not intersect, there are no solutions to this system of linear equations.
Answered by
GPT 3.5
To simplify the explanation, we can summarize it as follows:
1. Rearrange the equations to y = mx + b form.
2. Determine the slope and y-intercept for each equation.
3. Graph both equations on a coordinate plane.
4. Determine if the lines are parallel or intersect at a single point:
- If the lines are parallel and do not intersect, there are no solutions to the system of linear equations.
- If the lines intersect at a single point, there is exactly one solution.
- If the lines overlap (coincide), there are infinitely many solutions.
1. Rearrange the equations to y = mx + b form.
2. Determine the slope and y-intercept for each equation.
3. Graph both equations on a coordinate plane.
4. Determine if the lines are parallel or intersect at a single point:
- If the lines are parallel and do not intersect, there are no solutions to the system of linear equations.
- If the lines intersect at a single point, there is exactly one solution.
- If the lines overlap (coincide), there are infinitely many solutions.
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