19.01 Graphical method
State the number of solutions for each of the following graphed system of two linear equations:
Consider the following graphs of systems of linear equations:
How many solutions does this system of equations have?
Write down the solution(s) to the system of equations.
Consider the graph of the equation \\y = 3 x + 2:
If a second line intersects this line at the point \left(0, 2\right), can we determine the equation of the second line? Explain your answer.
Consider the graph of the equation \\y = 5 x + 3:
A second line, y = mx + b, intersects this line at the one point (0,3).
State the value of b. Explain your answer.
State the value that m cannot be equal to. Explain your answer.
A system of linear equations has no solutions. One of the equations of the system is \\y = - 3 x - 2. Determine whether the following could be the other equation of the system.
y = - 3 x - 3
y = \dfrac{x}{3} - 3
y = - \dfrac{x}{3} - 2
y = 3 x + 2
Use the given graph to solve each pair of simultaneous equations:
For each pair of linear equations:
Consider the following linear equations:
\begin{aligned} L_1: y &= - 2 x - 5 \\ L_2: y &= - 2 x + 4 \end{aligned}Sketch L_1 and L_2 on the same number plane.
Is there a coordinate that satisfies the two equations simultaneously? Explain your answer.
A rectangular zone is to be 3 \text{ m} longer than it is wide, with a total perimeter of 18 \text{ m}.
Let y represent the length of the rectangle and x represent the width. Construct two equations that represent this information.
Sketch the two lines on the same number plane.
Hence, find the length and width of the rectangle.
A band plans to record a demo at a local studio. The cost of renting studio A is \$250 plus \$50 per hour. The cost of renting studio B is \$50 plus \$100 per hour. The cost, y, in dollars of renting the studios for x hours can be modelled by the linear system:
Studio A: y = 50 x + 250
Studio B: y = 100 x + 50
Sketch the two lines on the same number plane.
State the coordinate which satisfies both equations.
What does the coordinate from part (d) mean?
Michael plans to start taking an aerobics class. Non-members pay \$4 per class. Members pay a \$10 one-time fee, but only have to pay \$2 per class. The monthly cost, y, of taking x classes can be modelled by the linear system:
Non-members: y = 4 x
Members: y = 2 x + 10
Sketch the two lines on the same number plane.
State the coordinate which satisfies both equations.
What does the coordinate from part (b) mean?
The cost of manufacturing toys, C, is related to the number of toys produced, n, by the formula C = 400 + 2 n. The revenue, R, made from selling n toys is given by R = 4 n.
Sketch the graphs of cost and revenue on the same number plane.
How many toys need to be produced for the revenue to equal the cost?
State the meaning of the y-coordinate of the point of intersection.
Given the cost function C \left( x \right) = 0.4 x + 2015 and the revenue function R \left( x \right) = 3 x, find the coordinates of the point of intersection, or the break-even point.
The two equations y = 3 x + 35 and y = 4 x represent Laura’s living expenses and income from work respectively.
Find the point of intersection of the two equations.
Sketch both equations on the same number plane.
State the meaning of the point of intersection of the two lines.
The two equations y = 4 x + 400 and y = 6 x represent a company's revenue and expenditure respectively.
Find the point of intersection of the two equations.
Sketch both equations on the same number plane.
State the meaning of the point of intersection of the two lines.