Write an equation that forms a system of equations with x + y = 4, so that the system has:
a) No solution b) Infinitely many solutions c) One solution
I know how to figure out if two linear systems have a solution, but I don't understand how you could figure out if just one linear system has a solution. Help please?
(a) no solution
recall that two linear equations have no solution if they are parallel (solution means they meet at a certain point),, thus the two equations, in order to be parallel must have the SAME SLOPE.
to determine the slope of the given equation, we rewrite it in this form:
y = mx + b
where m = slope and b = y-intercept,, so x + y = 4 becomes:
y = -x + 4
where m = -1
now that we know the slope, we can now formulate a new equation with slope=1,, thus we only change the value of b in order to have a parallel equation,, an example would be:
y = -x + 1
(b) infinitely many solutions
here, it means that the two linear equations coincide meaning they have the SAME SLOPE and Y-INTERCEPT,, this means that the equations are equal, and most of the time, they are just the multiple of each other.
for the given equation, we can just multiply any number to both sides of the equation to formulate the coinciding equation,, for instance we multiply it by 2:
2(x + y = 4)
2x + 2y = 8
(c) one solution
it means that the two equation meet at EXACTLY ONE POINT,, thus the other equation must have at least, a DIFFERENT SLOPE,, an example would be:
x + 2y = 4
where they meet at point (4,0)
hope this helps~ :)
sorry for the first post by the way~ ^^;
i just accidentally hit the enter button~ ^^;
(a) no solution
To determine if a linear system has a solution, we need to examine the relationship between the coefficients and constants in the equations. There are three possibilities: no solution, infinitely many solutions, or one solution.
a) No solution:
To create a system with no solution, we need two linear equations that are parallel and have different y-intercepts. In the given equation, x + y = 4, we can create another equation with the same slope but different y-intercept by multiplying both sides by a non-zero constant, such as 2(x + y) = 8. The resulting system would be:
x + y = 4
2x + 2y = 8
This system will have no solution since the lines are parallel and will never intersect.
b) Infinitely many solutions:
To create a system with infinitely many solutions, we need two linear equations that represent the same line. Continuing from the given equation, we can multiply both sides by the same non-zero constant, resulting in:
2(x + y) = 8
This simplifies to:
2x + 2y = 8
This system will have infinitely many solutions because the second equation is just a multiple of the first equation. The two equations represent the same line and intersect at every point on that line.
c) One solution:
For a system to have one unique solution, the two equations must represent two lines that intersect at a single point. Building upon the given equation x + y = 4, we can create a second equation that is not parallel or a multiple of the first equation. For example:
2x - y = 3
This system would have one solution since the two equations are not parallel and intersect at a single point.
To summarize, to determine the number of solutions in a linear system, we can manipulate the coefficients and constants of the equations to meet the criteria for each possibility.