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Two Equations, Two Unknowns
A pair of linear equations in two variables looks like a1x + b1y = c1 and a2x + b2y = c2. Each equation is a straight line on a graph, and the solution is the point where the lines meet. That picture gives the whole theory: two lines can cross at one point, never meet because they are parallel, or lie exactly on top of each other. Almost every word problem about two unknown quantities, such as the digits of a number, the parts of a fraction, the ages of two people or the prices of two items, becomes such a pair. Exams reward two skills: reading the nature of the solution without solving, and building the pair correctly from a sentence. This card set gives you a ratio test for the first and fixed setups for the second, with worked examples.
1. The pair 2x + 3y = 5 and 4x + 6y = 10 has:
2. For what value of k does the pair 2x + ky = 3 and 4x + 6y = 7 have no solution?
3. The digits of a two-digit number add up to 11, and the number exceeds its reverse by 9. The number is:
4. If 2x + 3y = 13 and 5x - 2y = 4, then the value of x times y is:
5. A fraction becomes 4/5 when 1 is added to both numerator and denominator, and 1/2 when 5 is subtracted from both. The fraction is:
6. A teacher draws two lines that intersect at exactly one point on the board. This shows that the pair of equations has:
7. A father is three times as old as his son. After 12 years he will be twice as old as the son. The father's present age is:
8. Which of the following pairs of equations has no solution?