Solutions to Equations with Two Variables
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A linear equation with two variablesAn equation with two variables that can be written in the standard form , where the real numbers a and b are not both zero. has standard form , where a, b, and c are real numbers and a and b are not both 0. Solutions to equations of this form are ordered pairs (x, y), where the coordinates, when substituted into the equation, produce a true statement.
Example 1: Determine whether (1, −2) and (−4, 1) are solutions to .
Solution: Substitute the x- and y-values into the equation to determine whether the ordered pair produces a true statement.
Answer: (1, −2) is a solution, and (−4, 1) is not.
It is often the case that a linear equation is given in a form where one of the variables, usually y, is isolated. If this is the case, then we can check that an ordered pair is a solution by substituting in a value for one of the coordinates and simplifying to see if we obtain the other.
Example 2: Are and solutions to ? Cleanmymac x 4 3 0 4. Receipts 1 7 4 – smart document collection job.
Solution: Substitute the x-values and simplify to see if the corresponding y-values are obtained.
Answer: is a solution, and is not.
Try this! Is (6, −1) a solution to ?
Answer: Yes
When given linear equations with two variables, we can solve for one of the variables, usually y, and obtain an equivalent equation as follows:
Written in this form, we can see that y depends on x. Here x is the independent variableThe variable that determines the values of other variables. Usually we think of the x-value as the independent variable. and y is the dependent variableThe variable whose value is determined by the value of the independent variable. Usually we think of the y-value as the dependent variable..
The linear equation can be used to find ordered pair solutions. If we substitute any real number for x, then we can simplify to find the corresponding y-value. For example, if , then , and we can form an ordered pair solution, (3, 2). Since there are infinitely many real numbers to choose for x, the linear equation has infinitely many ordered pair solutions (x, y).
Example 3: Find ordered pair solutions to the equation with the given x-values {−2, −1, 0, 4, 6}.
Solution: First, solve for y.
Next, substitute the x-values in the equation to find the corresponding y-values.
Answer: {(−2, −24), (−1, −19), (0, −14), (4, 6), (6, 16)}
In the previous example, certain x-values are given, but that is not always going to be the case. When treating x as the independent variable, we can choose any values for x and then substitute them into the equation to find the corresponding y-values. This method produces as many ordered pair solutions as we wish.
Example 4: Find five ordered pair solutions to .
Solution: First, solve for y.
Next, choose any set of x-values. Usually we choose some negative values and some positive values. In this case, we will find the corresponding y-values when x is {−2, −1, 0, 1, 2}. Make the substitutions required to fill in the following table (often referred to as a t-chart):
Answer: {(−2, 11), (−1, 8), (0, 5), (1, 2), (2, −1)}. Since there are infinitely many ordered pair solutions, answers may vary depending on the choice of values for the independent variable.
Try this! Find five ordered pair solutions to .
Answer: {(−2, −11), (−1, −6), (0, −1), (1, 4), (2, 9)} (answers may vary)
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