Questions: Describe the solution to the system of equations graphed below. The system has infinitely many solutions of the form y=-x+3, where x is any real number. The system has no solutions. The system has a unique solution at (2,1). The system has a unique solution at (1,2).

Solution Steps

The solution to a system of equations graphed is the point where the lines intersect. In this graph, the two lines intersect at the point (2,1).

The blue line passes through the points (0,3) and (3,0). The slope is given by \((0-3)/(3-0) = -1\). The y-intercept is 3. So the equation of the blue line is \(y = -x + 3\).

The red line passes through the points (1,3) and (2,1). The slope is given by \((1-3)/(2-1) = -2\). Using the point-slope form with the point (2,1), we get \(y - 1 = -2(x-2)\), which simplifies to \(y = -2x + 5\).

Substituting the point (2,1) in both equations: Blue line: \(1 = -2 + 3\), which simplifies to \(1 = 1\), so the point (2,1) lies on the blue line. Red line: \(1 = -2(2) + 5\), which simplifies to \(1 = 1\), so the point (2,1) lies on the red line.

Final Answer