3. Determine for what values of x the following expressions are negative. 4. Determine for what values of k the roots of the following equations are (a) real and unequal, (b) imaginary. CHAPTER X SIMULTANEOUS QUADRATIC EQUATIONS IN TWO 117. Solution of simultaneous quadratics. A single equation. in two variables, as x2 + y2 = 5, is satisfied by many pairs of values, as (1, 2), (√§, √§), (2, 1), and so on, though there are at the same time numberless pairs of values that do not satisfy it, as (0, 1), (1, 1), (2, 3). Thus the condition that (x, y) satisfy a single quadratic equation imposes a considerable restriction on the values that x and y may assume. If we further restrict the value of the pair of numbers (x, y) so that they also satisfy a second equation, the number of solutions is still further limited. The problem of solving two simultaneous equations consists in finding the pairs of numbers that satisfy them both. 118. Solution by substitution. In this method of solution the restriction imposed on (x, y) by one equation is imposed on the variables in the other equation by substitution. Solution: Equation (2) states that x = 1+y. Thus our desired solution is such a pair of numbers that (1) is satisfied and at the same time x is equal to y + 1. If we substitute in (1) 1+ y for x, we are imposing on its solution the restriction implied by (2). 119. Number of solutions. We have proved (p. 42) that two linear equations have in general one and only one solution. THEOREM. A quadratic equation and a linear equation have in general two and only two solutions. If the linear equation is solved for one variable, say x, and this is substituted in the quadratic equation, we get a quadratic equation to determine all possible values of the other variable (i.e. y), which must in general be two in number (§ 98). To each one of these values of y will correspond one and only one value of x, thus affording two solutions of the pair of equations. EXERCISES 1. When may, as a special case, a quadratic and a linear equation have only one solution? 2. When may a quadratic and a linear equation have imaginary solutions? 3. Find the real values of k for which the following equations have (1) only one solution, (2) imaginary solutions. 120. Solution when neither equation is linear. In the examples previously given one equation has been linear and the other quadratic in one or both variables. Often when neither of the original equations is linear a pair of equivalent (p. 41) equations one or both of which are linear may be found. These latter equations may be solved by substitution. EXERCISES Solve the following equations. When neither equation is linear, we can often obtain by addition an equation from which by the extraction of the square root a linear equation may be found. Thus the solutions are four in number, (4, 1), (1, 4), (— 1, — 4), (— 4, −1). The following exercise affords another case where a linear equation may be found by addition and extraction of the square root. Thus our solutions are (— 3, 5), (− 3, − 21), (3, + 21), (3, − 51). (1) (2) |