1.) Solve the system by graphing. You may use paper or a calculator.
2x - y = 8
x + 3y = 11
2.) The revenue for a certain brand of toothpaste is y = 2.5x where 'x' is the number of tubes of toothpaste sold and 'y' is the total income for selling 'x' tubes. The cost equation is y = 0.9x + 3000, where 'x' is the number of tubes manufactured and 'y' is the cost of producing 'x' tubes. Find the value of 'x' where the company breaks even.
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1.) To solve the system of equations by graphing, we will plot the lines represented by each equation on a graph and find the point(s) where the lines intersect.
The first equation is 2x - y = 8. Let's rearrange it to slope-intercept form, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
2x - y = 8
-y = -2x + 8
y = 2x - 8
The second equation is x + 3y = 11. Let's rearrange it to slope-intercept form:
x + 3y = 11
3y = -x + 11
y = (-1/3)x + 11/3
Now, plot these lines on a graph. Choose some x-values and calculate the corresponding y-values for each equation using the slope-intercept form. Then, plot the points and draw the lines through them.
Once the lines are plotted, find the point(s) where the lines intersect. These points represent the solution(s) to the system of equations.
2.) To find the value of 'x' where the company breaks even, we need to set the revenue equation equal to the cost equation and solve for 'x'.
The revenue equation is given as y = 2.5x, and the cost equation is y = 0.9x + 3000. We want to find the value of 'x' where the revenue equals the cost, so we can set the two equations equal to each other:
2.5x = 0.9x + 3000
Now, solve this equation for 'x' by isolating it on one side:
2.5x - 0.9x = 3000
1.6x = 3000
x = 3000 / 1.6
Use a calculator to evaluate the division:
x ≈ 1875
Therefore, the value of 'x' where the company breaks even is approximately 1875.