A system asks for values that make both equations true. When both equations give , their right sides must match at a solution. Set them equal and write the result as a quadratic equal to . If the question asks for the sum of its solutions, use the coefficients instead of finding each solution; keep the minus sign in the sum rule.
Hints
- Hint 1
A solution makes both equations true at once. Since each equation gives a value for , the two expressions on the right must be equal at every solution. What equation does that give you?
- Hint 2
Put the new equation into standard form, . Move the line’s -term first, then its constant, so you can identify the coefficient of without losing its sign.
- Hint 3
A root is an -value that makes a quadratic equal . In , the roots add to . The minus sign is part of the rule.
Step-by-step
Set the equations equal and use the root sum
Step 1Make the two expressions for y match
No Desmos needed. A coefficient rule will give the exact sum without finding either -value. At a solution, both equations give the same , so set their right sides equal:
- Step 2
Bring the x-terms together
Subtract from both sides:
- Step 3
Set the quadratic equal to zero
Add to both sides to get a quadratic, an equation with an term, equal to :
- Step 4
Find the sum from the coefficients
A root is an -value that makes this quadratic . If its roots are and , the -term of is . So for , the roots sum to . Here and . Keep both minus signs when you substitute:
The sum of all -values that satisfy the system is . Grid in 5.