Solve the system by graphing the lines. Express your answer as an ordered pair.
0.7x+0.8y=310
0.1x+0.4y=330
The solution is
The Correct Answer and Explanation is:
To solve the system of equations by graphing, we will first rewrite the given equations in slope-intercept form (i.e., (y = mx + b)), which is easier to graph. The given system of equations is:
- (0.7x + 0.8y = 310)
- (0.1x + 0.4y = 330)
Step 1: Solve the first equation for (y).
Start with:
[
0.7x + 0.8y = 310
]
To solve for (y), subtract (0.7x) from both sides:
[
0.8y = -0.7x + 310
]
Now, divide both sides by (0.8) to isolate (y):
[
y = \frac{-0.7x + 310}{0.8}
]
Simplify the expression:
[
y = -\frac{7}{8}x + 387.5
]
Now, the first equation is in slope-intercept form: (y = -\frac{7}{8}x + 387.5).
Step 2: Solve the second equation for (y).
Start with:
[
0.1x + 0.4y = 330
]
To solve for (y), subtract (0.1x) from both sides:
[
0.4y = -0.1x + 330
]
Now, divide both sides by (0.4) to isolate (y):
[
y = \frac{-0.1x + 330}{0.4}
]
Simplify the expression:
[
y = -\frac{1}{4}x + 825
]
Now, the second equation is in slope-intercept form: (y = -\frac{1}{4}x + 825).
Step 3: Graph the equations.
To graph these equations, we need the slope and the y-intercept for both.
For the first equation (y = -\frac{7}{8}x + 387.5), the slope is (-\frac{7}{8}) and the y-intercept is (387.5). This means the line crosses the y-axis at (387.5) and goes down 7 units for every 8 units it moves to the right.
For the second equation (y = -\frac{1}{4}x + 825), the slope is (-\frac{1}{4}) and the y-intercept is (825). This means the line crosses the y-axis at (825) and goes down 1 unit for every 4 units it moves to the right.
Step 4: Find the point of intersection.
To find the solution, we need to find the point where the two lines intersect. The point of intersection is the solution to the system of equations.
At this point, the values of (x) and (y) will satisfy both equations. To find the exact solution, we can use either substitution or elimination. However, since graphing these lines requires approximating the intersection visually, let’s proceed to find the exact coordinates algebraically.
Step 5: Solve algebraically (substitution or elimination method).
Let’s use substitution. Since we already have both equations in slope-intercept form:
- (y = -\frac{7}{8}x + 387.5)
- (y = -\frac{1}{4}x + 825)
Set the two equations equal to each other:
[
-\frac{7}{8}x + 387.5 = -\frac{1}{4}x + 825
]
Multiply everything by 8 to eliminate the fractions:
[
-7x + 3100 = -2x + 6600
]
Add (7x) to both sides:
[
3100 = 5x + 6600
]
Subtract 6600 from both sides:
[
-3500 = 5x
]
Divide by 5:
[
x = -700
]
Step 6: Solve for (y).
Now that we know (x = -700), substitute this value back into one of the original equations to solve for (y). Let’s use the second equation:
[
y = -\frac{1}{4}(-700) + 825
]
[
y = 175 + 825
]
[
y = 1000
]
Final Answer:
The solution to the system of equations is ((-700, 1000)).
Conclusion:
By solving the system algebraically and graphically, we find the solution to be ((-700, 1000)). The solution is the point where the two lines intersect on the graph. This method illustrates how graphing and algebraic manipulation can work together to find the solution.