The equation below involves the constant .
The equation has no solution. What is the value of ?
A linear equation that must have no solution needs its -terms to cancel while its remaining numbers disagree. Distribute, then match the coefficients, the factors multiplying , to find the constant. Finally, graph both sides in Desmos to check that they never meet. Matching coefficients alone can also leave an equation that every satisfies.
Hints
- Hint 1
For no solution, the -terms must cancel but the plain numbers must disagree. Use the distributive property on so you can see the -term on each side.
- Hint 2
A coefficient is the factor multiplying . The left coefficient is , and the right coefficient is . Set them equal so the -terms can cancel.
- Hint 3
Equal coefficients aren't the whole check. If the plain numbers match too, every works. Once you've found , put it into the original equation and see what remains.
Step-by-step
Match the coefficients, then check the constants
Step 1Distribute on the left
Use the distributive property: multiply by both and . The stays outside:
- Step 2
Make the x-terms cancel
The coefficients, or factors multiplying , must match for no solution. If they differ, you can solve for one . So set them equal:
After the -terms cancel, the remaining numbers must disagree.
- Step 3
Find what k + 3 must equal
Subtract from both sides:
Divide both sides by :
Don't divide by . It can equal , and dividing by would discard the value you need.
- Step 4
Check that no x works
Subtract from both sides:
Type , , and in Desmos. The graphs are the horizontal lines and . They never meet: the original equation becomes , which is false for every . Grid in -3.