Linda has a total of 225 on 3 tests. The sum of the scores on the first and second tests exceeds her third score by 61. Her first test exceeds her second by 6 points. Find Linda's 3 test scores.
Let the scores be x, y, z.
Solve the three simultaneous equations:
x + y + z = 225
x + y = z + 61
x = y + 6
Start by substitution: x=y+6
2 y + z = 219
2 y = z + 55
et cetera...
Let's assume Linda's scores on the three tests are x, y, and z, where x is her first score, y is her second score, and z is her third score.
According to the given information:
1) x + y + z = 225 (the total of all three scores is 225).
2) x + y = z + 61 (the sum of the first and second scores exceeds her third score by 61).
3) x = y + 6 (her first score exceeds her second score by 6 points).
To find the scores, we need to solve these three equations simultaneously.
Let's start by substituting the third equation into the second equation:
(y + 6) + y = z + 61
2y + 6 = z + 61
2y = z + 55
z = 2y - 55
Now, substitute this expression for z back into the first equation:
x + y + (2y - 55) = 225
x + 3y - 55 = 225
x + 3y = 280
We have reduced the problem to two equations:
x + 3y = 280 ...(Equation 4)
2y - 55 = z ...(Equation 5)
Now, we need to find values of x, y, and z that satisfy equations 4 and 5.
To further simplify the problem, let's solve equation 4 for x:
x = 280 - 3y
Now, substitute this expression for x into equation 5:
2y - 55 = 280 - 3y
Simplify the equation:
2y + 3y = 280 + 55
5y = 335
y = 335/5
y = 67
Now, substitute y = 67 back into equation 4:
x + 3(67) = 280
x + 201 = 280
x = 280 - 201
x = 79
Finally, substitute the values of x = 79 and y = 67 back into equation 5 to find z:
2(67) - 55 = z
134 - 55 = z
z = 79
Therefore, Linda's test scores are x = 79, y = 67, and z = 79.