Which ordered pair is a solution of the equation? x+7y=17
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To determine if the ordered pair (3, 2) is a solution to the equation x plus 7y equals 17, we substitute the values into the equation. When we substitute x equals 3 and y equals 2, we get 3 plus 7 times 2, which equals 3 plus 14, which equals 17. Since the left side equals the right side of the equation, 17 equals 17, the ordered pair (3, 2) is indeed a solution to the equation. We can also see this graphically, as the point (3, 2) lies on the line representing the equation x plus 7y equals 17.
Let's check if other ordered pairs are solutions to the equation x plus 7y equals 17. First, let's try the point (10, 1). Substituting x equals 10 and y equals 1 into the equation, we get 10 plus 7 times 1, which equals 10 plus 7, which equals 17. Since 17 equals 17, the point (10, 1) is also a solution. Now, let's try the point (3, 3). Substituting x equals 3 and y equals 3, we get 3 plus 7 times 3, which equals 3 plus 21, which equals 24. Since 24 does not equal 17, the point (3, 3) is not a solution. On our graph, we can see that the points (3, 2) and (10, 1) lie on the line, while (3, 3) does not.
Let's explore how to find solutions to the equation x plus 7y equals 17. We can choose any value for one variable and then solve for the other. For example, if we set x equal to 0, we get 0 plus 7y equals 17, which simplifies to 7y equals 17. Solving for y, we get y equals 17 divided by 7, which is approximately 2.43. So the point (0, 2.43) is a solution. Similarly, if we set y equal to 0, we get x plus 7 times 0 equals 17, which simplifies to x equals 17. So the point (17, 0) is also a solution. These points are the y-intercept and x-intercept of the line, respectively. In fact, there are infinitely many solutions to this equation, and they all lie on the line x plus 7y equals 17. We've already verified that (3, 2) and (10, 1) are also solutions.
Let's solve the equation x plus 7y equals 17 for y in terms of x. First, we subtract x from both sides to get 7y equals 17 minus x. Then, we divide both sides by 7 to isolate y, giving us y equals 17 minus x, all divided by 7. This can be rewritten in slope-intercept form as y equals negative one-seventh x plus seventeen-sevenths. From this form, we can identify the slope as negative one-seventh and the y-intercept as seventeen-sevenths, which is approximately 2.43. On our graph, we can visualize the slope by moving 7 units to the right and 1 unit down from the y-intercept. This gives us a slope of negative one-seventh, which means for every 7 units we move horizontally, we move down 1 unit vertically.
Let's summarize what we've learned about the equation x plus 7y equals 17. To check if an ordered pair is a solution, we substitute the values into the equation and verify that both sides are equal. We confirmed that the ordered pairs (3, 2) and (10, 1) are solutions because they satisfy the equation. We also found that the equation can be rewritten in slope-intercept form as y equals negative one-seventh x plus seventeen-sevenths. From this form, we identified that the line has a slope of negative one-seventh and a y-intercept of approximately 2.43. Most importantly, we learned that there are infinitely many solutions to this equation, and all these solutions lie on the line x plus 7y equals 17. Therefore, the ordered pair (3, 2) is indeed a solution to the equation x plus 7y equals 17.