A system of equations asks for points that satisfy both equations. When a line and a parabola are given, type both equations into Desmos and click every intersection. Each intersection is a solution ; read its second coordinate, then add the -coordinates. Adding the first coordinates instead answers a different question.
Hints
- Hint 1
A solution to a system lies on both graphs, so it appears where the line and parabola cross. Look for both crossing points, not where either graph crosses an axis.
- Hint 2
A point is written , so is the second number. After you click both intersections, which two second numbers should you add?
Step-by-step
Approach 1: Read both intersections in Desmos
- Step 1
Find the two solutions
Type and on separate lines. A solution lies on both graphs, so click both intersections. Desmos shows and .
- Step 2
Add the requested coordinates
In each point , the second number is . Take the -coordinate from each solution, not the -coordinate. Add from and from :
The sum of the solutions' -coordinates is . Grid in 4.
Approach 2: Use the sum of the roots
- Step 1
Make one equation for the shared x-values
At each solution, both equations give the same , so set their right sides equal:
- Step 2
Put the quadratic in standard form
Subtract from both sides:
Combine like terms:
- Step 3
Find the sum of the x-values
The roots are the -values that make this quadratic equal . For , their sum is . Here and , so keep the minus sign:
- Step 4
Turn the x-sum into a y-sum
The line says , so each -coordinate is less than its -coordinate. Add the two line equations:
Group the -values:
Use the sum you found:
Subtract:
The sum of the -coordinates is . Grid in 4.