Sign In
| 11 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Maya is researching a toy factory that currently produces robot action figures and car toys.
In a system of equations, an equivalent system can be created by replacing one equation with the sum of two or more equations in the system or by replacing an equation with a multiple of itself. An equation can be also replaced by the sum of that equation and a multiple of another equation in the system.
Remove parentheses
Commutative Property of Addition
Add and subtract terms
For each system of linear equations, verify whether the coordinate pair is a solution.
A vlogger that Diego likes to watch bought silver and gold to make Olympic-style medals. The vlogger will show the process of making the medals as a multi-video series.
(I): LHS/25=RHS/25
(I): Write as a sum of fractions
(II): LHS/5=RHS/5
(II): Write as a difference of fractions
(I), (II): ca⋅b=ca⋅b
(I), (II): Calculate quotient
(I), (II): Identity Property of Multiplication
(I): LHS−80g=RHS−80g
(II): LHS+10g=RHS+10g
Looking at the graph, it can be seen that the solution is about 3 ounces of gold and a bit more than 40 ounces of silver. In this case, since the point of intersection is not a lattice point, finding the exact solution is not possible.
(II): LHS/5=RHS/5
(II): Write as a difference of fractions
(II): ca⋅b=ca⋅b
(II): Calculate quotient
(II): Identity Property of Multiplication
(II): LHS+10g=RHS+10g
(I): s=10g+13
(I): LHS/5=RHS/5
(I): Write as a sum of fractions
(I): ca⋅b=ca⋅b
(I): Calculate quotient
(I): Subtract (II)
(I): Distribute -1
(I): Add and subtract terms
(I): LHS/450=RHS/450
Concept | Definition |
---|---|
Consistent System | A system of equations that has at least one solution. |
Inconsistent System | A system of equations that has no solution. |
Dependent System | A system of equations with infinitely many solutions. |
Independent System | A system of equations that has exactly one solution. |
Since the system has exactly one solution, it is both consistent and independent.
As part of his student council duties, Tadeo is in charge of getting equipment for the school gym. At the moment, the gym needs new basketballs and volleyballs. Each basketball costs $75 and each volleyball costs $50. The gym's budget is $1400.
(I): LHS/(-5)=RHS/(-5)
(I): Write as a sum of fractions
(I): ca⋅b=ca⋅b
(I): Put minus sign in front of fraction
a+(-b)=a−b
(I): Calculate quotient
(I): Add (II)
(I): Remove parentheses
(I): Add and subtract terms
(I): LHS/7=RHS/7
Equation (I) | Equation (II) | |
---|---|---|
Equation | 75b+50v=1400 | 22b+10v=364 |
Substitute | 75(12)+50(10)=1400 | 22(12)+10(10)=364 |
Simplify | 1400=1400 ✓ | 364=364 ✓ |
The values verify both equations of the system. Therefore, the solution is correct.
Concept | Definition |
---|---|
Consistent System | A system of equations that has at least one solution. |
Inconsistent System | A system of equations that has no solution. |
Dependent System | A system of equations with infinitely many solutions. |
Independent System | A system of equations that has exactly one solution. |
Since the system has exactly one solution, the system is both consistent and independent.
Ignacio is looking through his dad's toolbox to see if there are any duplicates that can be sold at a garage sale. He discovers that there are some extra hammers and wrenches.
(I): LHS⋅3.5=RHS⋅3.5
(I): Distribute 3.5
(II): Subtract (I)
(II): Distribute -1
(II): Subtract terms
Concept | Definition |
---|---|
Consistent System | A system of equations that has at least one solution. |
Inconsistent System | A system of equations that has no solution. |
Dependent System | A system of equations with infinitely many solutions. |
Independent System | A system of equations that has exactly one solution. |
Since the system has infinitely many solutions, the system is consistent and dependent. In a system of linear equations, if the system is dependent, then the equations that make the system are equations of coincidental lines.
Concept | Definition |
---|---|
Consistent System | A system of equations that has at least one solution. |
Inconsistent System | A system of equations that has no solution. |
Dependent System | A system of equations with infinitely many solutions. |
Independent System | A system of equations that has exactly one solution. |
Since the system has no solution, it is an inconsistent system.
Solve the system of linear equations to find the values of x and y.
Multiply or divide an equation by a number so that a variable has the same coefficient in both equations.
(I): LHS⋅10=RHS⋅10
(I): Distribute 10
(I): Commutative Property of Multiplication
(I): ca⋅b=ca⋅b
(I): Calculate quotient
(II): LHS/5=RHS/5
(II): Write as a sum of fractions
(II): ca⋅b=ca⋅b
(II): Calculate quotient
(I), (II): Identity Property of Multiplication