Parallel Lines and Inconsistent Systems

When Two Lines Cannot Meet

A system of two linear equations in two variables asks for an ordered pair \((x,y)\) that satisfies both equations. Geometrically, that pair would be a point lying on both lines. When two lines have the same direction but are distinct, they never share a point, so their equations form an inconsistent system. The key comparison for nonvertical lines is between their slopes and their intercepts: equal slopes give the same direction, while different intercepts keep the lines apart.

A system is consistent if at least one ordered pair satisfies all its equations, and inconsistent if no such pair exists. A nonvertical line can be written in slope-intercept form \(y=mx+b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Every point \((x,y)\) on the line satisfies this equation. For an equation \(Ax+By=C\) with \(B\ne0\), solving for \(y\) gives \(y=-\frac{A}{B}x+\frac{C}{B}\). Thus its slope is \(-\frac{A}{B}\), and its \(y\)-intercept is \(\frac{C}{B}\).

It is important to compare both parts of the equations. Two nonvertical lines with equal slopes and equal intercepts are the very same line, not distinct parallel lines. Equal slopes together with different intercepts describe distinct parallel lines, and that difference is precisely what prevents a common point.

The Slope-and-Intercept Test

Proof. First suppose \(m_1=m_2\) and \(b_1\ne b_2\). If a point \((x,y)\) lay on both lines, it would satisfy \(y=m_1x+b_1\) and \(y=m_2x+b_2\). Equating the right sides and using \(m_1=m_2\) gives \(m_1x+b_1=m_1x+b_2\), hence \(b_1=b_2\), a contradiction. So there is no common point. The lines have the same slope, hence the same direction, and the absence of a common point means they are distinct parallel lines. Conversely, suppose the lines are distinct and parallel. If \(m_1\ne m_2\), setting their expressions for \(y\) equal gives \((m_1-m_2)x=b_2-b_1\), which has a real solution \(x=\frac{b_2-b_1}{m_1-m_2}\). Taking \(y=m_1x+b_1\) then gives a point on both lines, contradicting that they are parallel and distinct. Therefore \(m_1=m_2\). If \(b_1=b_2\) as well, the equations are identical, so the lines coincide, contradicting distinctness. Thus \(b_1\ne b_2\). Since a solution to the system would be a common point, the system is inconsistent. \(\square\)

The proof also explains why slope alone is not enough to distinguish parallel lines from coincident lines. If both the slope and intercept match, every point on one line satisfies the other equation. If the slopes differ, the equality of the two expressions for \(y\) determines a common point. Only equal slopes with unequal intercepts rule out a common point for nonvertical lines.

Vertical Lines

The slope-intercept form \(y=mx+b\) cannot represent a vertical line, because a vertical line has a fixed \(x\)-coordinate and does not give one value of \(y\) for each \(x\). Its equation is \(x=a\). Two vertical lines \(x=a\) and \(x=d\) have the same direction. If \(a=d\), they coincide; if \(a\ne d\), they cannot share a point, because a shared point would need its \(x\)-coordinate to equal both \(a\) and \(d\).

Proof. If \(a\ne d\), any solution \((x,y)\) would have to satisfy \(x=a\) and \(x=d\), forcing \(a=d\), a contradiction. Hence the system has no solution. If \(a=d\), the equations are identical. For any real \(y\), the point \((a,y)\) satisfies both, so the system is consistent and the lines coincide. \(\square\)

A vertical line and a nonvertical line are different from the equal-slope case: they do not have the same direction. In fact, if the vertical line is \(x=a\) and the nonvertical line is \(y=mx+b\), substituting \(x=a\) gives \(y=ma+b\). Thus \((a,ma+b)\) lies on both. This is why a vertical line should be handled by comparing its fixed \(x\)-coordinate, not by trying to assign it a finite slope.

Reading the Test from Standard Form

In standard form \(Ax+By=C\), with \(A\) and \(B\) not both zero, the value of \(B\) tells us which comparison to make. When \(B\ne0\), divide by \(B\) and solve for \(y\) to find the slope and intercept. When \(B=0\), the equation becomes \(Ax=C\). Since \(A\ne0\), this is the vertical line \(x=\frac{C}{A}\). For two equations, compare their slopes and intercepts if both are nonvertical; if both are vertical, compare their fixed \(x\)-coordinates.

Common Mistake: Matching Coefficients but Ignoring Constants

A common error is to see the same coefficients of \(x\) and \(y\) in two equations and conclude that they describe the same line. Consider \(2x+3y=6\) and \(2x+3y=9\). The incorrect conclusion is that the equations are identical because their left sides match. But the right sides differ, so the equations impose different constraints.

Solving each for \(y\), the first gives \(3y=6-2x\), hence \(y=-\frac{2}{3}x+2\). The second gives \(3y=9-2x\), hence \(y=-\frac{2}{3}x+3\). The slopes are equal and the intercepts differ, so the lines are distinct and parallel. For a direct check, the point \((0,2)\) satisfies the first equation because \(2(0)+3(2)=6\), but it does not satisfy the second because \(2(0)+3(2)=6\ne9\). More generally, a common solution would force the same quantity \(2x+3y\) to equal both \(6\) and \(9\), which is impossible.

When testing two equations, keep the constant terms in the comparison. For nonvertical lines, compare both slope and intercept; for vertical lines, compare the fixed \(x\)-coordinates. In either case, the claim of inconsistency rests on showing that a common point would have to satisfy incompatible requirements.