Reasoning Without graphing, decide whether the following system of linear equations has one solution, infinitely many solutions, or no solution. Explain.
To decide whether the system of linear equations has one solution, infinitely many solutions, or no solution without graphing, we can use the method of elimination or substitution.
If the system has one solution, it means that the two equations represent two distinct lines that intersect at a single point.
If the system has infinitely many solutions, it means that the two equations represent two distinct lines that are coincident or overlapping. In other words, the two equations are essentially the same line.
If the system has no solution, it means that the two equations represent two distinct lines that are parallel and do not intersect.
Now, let's consider the system of linear equations without graphing:
Equation 1: 4x - 2y = 8
Equation 2: 2x - y = 4
To determine the solution(s), we can use the method of substitution:
From Equation 2, we can isolate y:
y = 2x - 4
Substitute this expression into Equation 1:
4x - 2(2x - 4) = 8
4x - 4x + 8 = 8
8 = 8
The equation simplifies to 8 = 8. This equation is always true, regardless of the value of x. In other words, any value of x will satisfy the equation.
Since any value of x satisfies the equation, we have infinitely many solutions. This means that the system of linear equations has infinitely many solutions.
Therefore, without graphing, we can conclude that the system of linear equations has infinitely many solutions.