Substitution Method
Consider the following word problem:
Eleanor is working her way through school. She works two part-time jobs for a total of 20 hours a week. Job A pays $\$ 6.20$ per hour, and Job B pays $\$ 7.40$ per hou How many hours did she work at each job the week that she made $\$ 134.80$.
Step 1 of 2: Use the variables $x$ and $y$ to set up two equations to solve the given problem.
Answer
First Equation:
Second Equation:
Submit
Addition/Elimination Method
Solve for the variable x in the system of equations:
$\begin{array}{l}4 x+z=4 \\ x-y+4 z=-11 \\ -3 x+y-5 z=11\end{array}$
Graphing Method
Determine the solution region for the following system of linear inqeualities by inputting a point in that region. The graph, without shaded solution, is shown on the right.
(Enter an ordered pair $(x, y)$ with integer values, that is, no decimals or fractions.) (Do not enter a point that is actually on one of the lines; keep it in the interior of the solution region.)
\[
\begin{array}{r}
-3 x-3 y<24 \\
-3 x+y<12
\end{array}
\]
Point in the Solution Region:
Determining if the Point is a Solution
Decide whether the given ordered pair is a solution of the given system.
\[
\begin{array}{c}
(5,-7) \\
x+y=-2 \\
2 x+3 y=16
\end{array}
\]
Is the s:dered pair a solution to the system of equations?
Determining Parallel Lines
System A
Line 1: \( y=\frac{5}{2} x+3 \)
Line \( 2: y=\frac{5}{2} x-3 \)
The system has exactly one solution.
Solution: \( \square, \square) \)
The system has infinitely many solutions.
The system has no solution.
Determining Perpendicular Lines
Given the equations of two lines, \(y = 2x + 3\) and \(y = mx + 4\), find the value of \(m\) that will make the two lines perpendicular.
Dependent, Independent, and Inconsistent Systems
Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $\$ 22$ and a charge of $\$ 0.06$ per minute for calls. Company B has a monthly fee of $\$ 6$ and a charge of $\$ 0.14$ per minute for calls.
How many minutes of calling would make the two plans equal?
Answer: $\mid$ (Enter a numeric response, include the correct units )
Finding the Intersection (and)
Use the graph that shows the solution to $f(x)=g(x)$
\[
\begin{array}{l}
f(x)=\frac{7}{3} x-3 \\
g(x)=2^{x}-4
\end{array}
\]
What is the solution to $f(x)=g(x)$ ?
Select each correct answer.
Using the Simplex Method for Constraint Maximization
Solve the following system of linear inequalities using the simplex method and find the maximum value of the objective function: \[ \begin{cases} x + y \leq 5 \\ 2x + y \leq 8 \\ x, y \geq 0 \end{cases} \] where the objective function is \( Z = 2x + 3y \).
Using the Simplex Method for Constraint Minimization
Minimize the objective function P = 2x + 3y subject to the following constraints:\[ \begin{align*} x + y &\geq 3 \\ 2x + y &\leq 6 \\ x, y &\geq 0 \end{align*} \] using the Simplex Method.
Finding the Union (or)
Solve the following system of equations and find the union of the solutions: \(2x + 3y = 6\) and \(5x - y = 10\)
Finding the Equation with Real Coefficients
Given that the system of equations $2x + y = 5$ and $3x - 2y = 7$ has a solution, find the equation of the line parallel to the line whose equation is formed by eliminating $y$ from these two equations.
Finding a Direct Variation Equation
23.甲、乙二人练习跑步, 若甲让乙先跑10米, 则甲跑5秒可追上乙; 若乙比甲先跑2秒, 则甲跑4秒能追上 乙。问: 两人每秒各跑多少米?
解: 甲乙速度差为 $10 / 5=2$
Finding the Slope for Every Equation
Given the system of equations: \(3x - 2y = 2\) and \(6x + 4y = 8\), find the slope for each equation.
Evaluating
Solve the system of equations \(2x + 3y = 12\) and \(5x - y = 9\).
Finding a Variable Using the Constant of Variation
A system of equations is given by \(y = kx\) and \(y = 3x + 5\) where \(k\) is a constant. Find the value of \(k\) if the system has a unique solution.