A system of linear equations is given by the tables. One of the tables is represented by the equation [tex]y=-\frac{1}{3} x+7[/tex].
\begin{tabular}{|c|c|} \hline$x$ & $y$ \ \hline 0 & 5 \ \hline 3 & 6 \ \hline 6 & 7 \ \hline 9 & 8 \ \hline \end{tabular}
\begin{tabular}{|c|c|} \hline $x$ & $y$ \ \hline-6 & 9 \ \hline-3 & 8 \ \hline 0 & 7 \ \hline 3 & 6 \ \hline \end{tabular}
The equation that represents the other equation is [tex]y=[/tex] $\square$ [tex]x+[/tex] $\square$
The solution of the system is ($\square$ , $\square$)
Answer (2)
Determine the equation for the first table: y = 3 1 x + 5 .
Set the two equations equal to each other: − 3 1 x + 7 = 3 1 x + 5 .
Solve for x : x = 3 .
Substitute x = 3 into either equation to find y : y = 6 . The solution is ( 3 , 6 ) .
Explanation
Problem Analysis We are given a system of linear equations represented by two tables. One of the equations is y = − 3 1 x + 7 . Our goal is to find the other equation and the solution to the system.
Identifying the Equation for Table 2 First, let's determine which table corresponds to the equation y = − 3 1 x + 7 . We can test the points in each table. Table 2 contains the points (-6, 9), (-3, 8), (0, 7), (3, 6). If we plug these x values into the equation, we get:
For x = -6: y = − 3 1 ( − 6 ) + 7 = 2 + 7 = 9 For x = -3: y = − 3 1 ( − 3 ) + 7 = 1 + 7 = 8 For x = 0: y = − 3 1 ( 0 ) + 7 = 0 + 7 = 7 For x = 3: y = − 3 1 ( 3 ) + 7 = − 1 + 7 = 6
Since all the points in Table 2 satisfy the equation y = − 3 1 x + 7 , Table 2 corresponds to this equation.
Finding the Equation for Table 1 Now, let's find the equation corresponding to Table 1, which contains the points (0, 5), (3, 6), (6, 7), (9, 8). We can use the points (0, 5) and (3, 6) to find the slope and y-intercept.
The slope m is given by m = x 2 − x 1 y 2 − y 1 . Using the points (0, 5) and (3, 6), we have
m = 3 − 0 6 − 5 = 3 1
Since the point (0, 5) is given, the y-intercept is 5. Thus, the equation is y = 3 1 x + 5 .
Solving the System of Equations Next, we solve the system of equations:
y = − 3 1 x + 7 y = 3 1 x + 5
Setting the two equations equal to each other:
− 3 1 x + 7 = 3 1 x + 5
Solving for x :
3 2 x = 2 x = 3
Substituting x = 3 into either equation to find y . Using the second equation:
y = 3 1 ( 3 ) + 5 = 1 + 5 = 6
Thus, the solution is (3, 6).
Final Answer The equation that represents the other equation is y = 3 1 x + 5 , and the solution of the system is (3, 6).
Examples
Imagine you're tracking the growth of two plants. Plant A starts at 5 inches and grows 1/3 inch per day, while Plant B starts at 7 inches and shrinks 1/3 inch per day. This problem helps you determine when the plants will be the same height and what that height will be. Understanding systems of equations can guide you in scenarios such as comparing growth rates, predicting intersection points, or optimizing resource allocation. This algebraic approach ensures accurate comparisons and predictions in practical tasks.
The other equation obtained from Table 1 is y = 3 1 x + 5 , and the solution to the system of linear equations is ( 3 , 6 ) .
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