Lesson 4: Equations with No Solutions or Infinitely Many Solutions

Property

The result of this process may be “all numbers” or “a particular number” or “no numbers”.
These occur when the coefficient of $x$ is the same on both sides of the equation.

Examples

• One solution: $5x - 2 = 3x + 8$ simplifies to $2x = 10$, so $x = 5$ is the single solution.

• No solution: $3(x+2) = 3x - 1$ becomes $3x+6 = 3x-1$. Subtracting $3x$ from both sides gives the false statement $6 = -1$, so there is no solution.

Property

Steps for Solving Linear Equations.

1. Use the distributive law to remove any parentheses.
2. Combine like terms on each side of the equation.
3. By adding or subtracting the same quantity on both sides of the equation, get all the variable terms on one side and all the constant terms on the other.
4. Divide both sides by the coefficient of the variable to obtain an equation of the form $x = a$.

Examples

• Solve $3(x-4) = 9$. First, distribute the 3 to get $3x - 12 = 9$. Add 12 to both sides to get $3x = 21$. Finally, divide by 3 to find $x = 7$.
• Solve $5(y+1) = 2y - 4$. Distribute to get $5y+5 = 2y-4$. Subtract $2y$ from both sides, then subtract 5 from both sides to get $3y = -9$. Divide by 3 to find $y=-3$.
• Solve $25 - 4x = 2x - 5(2-x)$. Distribute to get $25 - 4x = 2x - 10 + 5x$. Combine like terms to get $25 - 4x = 7x - 10$. Add $4x$ to both sides, then add 10 to both sides to get $35 = 11x$, so $x = \frac{35}{11}$.

Explanation

When an equation has parentheses, first use the distributive law to clear them. After that, tidy up by combining like terms on each side. This simplifies the equation, making it easier to isolate the variable and find your solution.