Systems from Word Problems - Algebra Step by Step

You buy 7 items at a shop: some cost $1 each and some cost $2 each. The total bill is $9. How many of each did you buy? Let a = number of $1 items, r = number of $2 items. Constraints: a + r = 7 (count) and a + 2r = 9 (cost in dollars).

word-problem-system Model context produces the equations; the transcript solves them. Identify unknowns, write one equation per constraint, then apply elimination. Verify both equations with the solution.

Example

Translate constraints into equations, then solve the system. Each condition in the problem becomes one equation relating the unknowns, and together they pin down values that satisfy all conditions at once. Solving the system finds the point where the constraints agree, which is why a well-posed problem needs as many independent equations as unknowns.

highlighted = computed this step

Step 1 — Translate

Set up the expression.

a+r= 7 ,\ a+ 2r = 9

Step 2 — Subtract equations

Subtract equations: 9 - 7 = 2.

r= 9 - 7 = \hl{2}

Step 3 — Back-substitute

Back-substitute: 7 - 2 = 5.

a= 7 - 2 = \hl{5}

Step 4 — Solution

Read the final result.

a= 5 ,\ r= 2