To find the best estimate for the solution of the system of linear equations represented by the two lines, we can analyze the equations given:
- \( x + y = 975 \) (which is the total amount invested)
- \( 0.05x + 0.12y = 111 \) (which represents the total interest earned)
We want to find the point where these two lines intersect, which represents the values of \( x \) (amount invested at 5%) and \( y \) (amount invested at 12%).
Looking at the points provided:
- (85, 900)
- (900, 85)
- (880, 120)
- (120, 880)
None of these points explicitly represent the values of \( x \) and \( y \) that will solve the equations perfectly without solving the equations directly. However, we can estimate based on the total investment \( x + y = 975 \).
Knowing that one of the investments is significantly higher in percentage return (12%), we expect more money to be skewed toward that investment when total interest is considered.
Now let's take a look mathematically at the options:
If we were to assume \( x = 900 \) and \( y = 85 \), that would imply:
- Total: \( 900 + 85 = 985 \) which is incorrect as it exceeds 975
- Total interest: \( 0.05(900) + 0.12(85) = 45 + 10.2 = 55.2 \) which is under 111.
Continuing this way for each of the points would lead us to see that:
(880, 120):
- Total: \( 880 + 120 = 1000 \) which is incorrect as it exceeds 975.
- Total interest: \( 0.05(880) + 0.12(120) = 44 + 14.4 = 58.4 \) also under.
(120, 880):
- Total: \( 120 + 880 = 1000 \) again over.
Finally checking point (85, 900):
- Total: \( 85 + 900 = 985 \) also too high.
Upon further inspection, we notice that some incorrect values exist within the assumed points indicating that manual checking needs to be conducted;
Continuing this analysis of carefully substituting values into both equations, we realize that if we adjust and solve completely: The right approach must involve some more targeted values or a possible visual method to represent and find out where the plot visually intersects rather than relying on potential conflicts we see in verbal math naturally.
Thus, for estimations and the graphical intersection — the best one leading us in the possible sense must fall toward: Best estimate point to represent the solution of the equations is: (900, 85).
Inferring both potential arguments for success from both equations where \(y\) might edge toward total claim on return recognition thus leading us back to: (900, 85) suggested.
Final Response: The point that is the best estimate to the solution of the system of linear equations is (900, 85).
1 answer