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Consider the equations you circled as having one solution.

What does it mean if an equation has one solution?
(Please be clear and thorough with your response.)

Consider the equations you circled as having infinite solutions.
What does it mean if an equation has infinite solutions?
(Please be clear and thorough with your response.)

1 answer

One Solution

When we say an equation has one solution, it means that there is exactly one unique value for the variable that satisfies the equation. In other words, if you were to substitute this value back into the equation, both sides of the equation would be equal.

For example, consider the linear equation \(2x + 3 = 7\). To solve for \(x\), we would isolate \(x\):

  1. Subtract 3 from both sides: \[ 2x = 4 \]
  2. Divide both sides by 2: \[ x = 2 \]

In this case, the equation has one solution, \(x = 2\). This indicates that, geometrically, if you were to graph the equation, it would represent a line that intersects the x-axis at only one point, which means there is only one unique point (or solution) where this equation holds true.

Infinite Solutions

In contrast, when an equation has infinite solutions, it means that there are endless values for the variable that satisfy the equation. This typically occurs when two equations represent the same line or when one equation can be derived from another through algebraic manipulation.

For instance, consider the equation \(2x + 4 = 8\). If we simplify:

  1. Subtract 4 from both sides: \[ 2x = 4 \]
  2. Divide both sides by 2: \[ x = 2 \]

Now, if we have another equation represented as \(x - 2 = 0\), both equations simplify to the same solution, \(x = 2\). However, if we look at a situation where we have two equations like \(2x + 4 = 8\) and \(4x + 8 = 16\), we can simplify both equations to the same form:

  • The first simplifies to \(x = 2\).
  • The second simplifies to \(x = 2\).

In this case, directly manipulating the second equation shows that they yield the same definition for \(x\), resulting in an infinite number of solutions along the same line.

Geometrically, when graphing, an equation with infinite solutions would be represented as a line that overlaps with another line entirely, meaning they have all points in common. Hence, instead of intersecting at just one point, they coincide at every point along that line.

Summary

  • One Solution: There is a single unique value for the variable, leading to one specific point where the equation holds true.
  • Infinite Solutions: There are countless values for the variable that satisfy the equation, representing a situation where the equations describe the same geometric entity (like a line) in an overlapping manner.