Let x be the number of five-point problems, and y the number of two-point problems. Create and solve a system of equations.
The test has 8 five-point problems and 30 two-point problems.
Practice makes perfect
In order to find the solution to the given problem, we will write two equations and solve the system formed.
Writing the System
Let x be the number of five-point problems and y the number of two-point problems. Knowing that the test consists of 38 problems, the sum of x and y must be 38.
x+y=38
Let's now write our second equation. The expression 5x represents the number of points obtained from five-point questions, and 2y represents the points obtained from two-point questions. The test is worth 100 points. Therefore, the sum of these two expressions must equal 100.
5x+2y=100
With our two equations, we can form a system.
x+y=38 & (I) 5x+2y=100 & (II)
Solving the System
Note that the y-variable in Equation (I) can be isolated in just one step.
x+y=38 5x+2y=100 ⇔ y=38-x & (I) 5x+2y=100 & (II)
Since one variable is isolated, we will use the Substitution Method.
The solution to the system, which is the point of intersection of the line, is (8,30). In the context of the problem, this means that there are 8 five-point problems and 30 two-point problems in the test.
