{{ item.displayTitle }}

No history yet!

Student

Teacher

{{ item.displayTitle }}

{{ item.subject.displayTitle }}

{{ searchError }}

{{ courseTrack.displayTitle }} {{ statistics.percent }}% Sign in to view progress

{{ printedBook.courseTrack.name }} {{ printedBook.name }} a

We want to find the number of solutions for the following system of equations. ${-21 y=23 x−42y+6x=6 (I)(II) $ To determine how many solutions this system has, we will solve it by substitution. Doing so will result in one of three cases.

Result of solving by substitution | Number of solutions |
---|---|

A value for $x$ and $y$ is determined. | One solution |

An identity is found, such as $2=2.$ | Infinitely many solutions |

A contradiction is found, such as $2 =3.$ | No solution |

Thus, we need to solve the system of equations before we can make our conclusion. When solving a system of equations using substitution, there are three steps.

- Isolate a variable in one of the equations.
- Substitute the expression for that variable into the other equation and solve.
- Substitute this solution into one of the equations and solve for the value of the other variable.

$2y+6x=6$

Substitute$y=-3x+8$

$2(-3x+8)+6x=6$

DistrDistribute $2$

$-6x+16+6x=6$

AddTermsAdd terms

$16 =6$

b

Here we have been asked to determine how many solutions the system ${2y=-x+22y−5x=0 (I)(II) $ has. We will first solve the system by substitution. After that we will use the following table to find the number of solutions.

Result of solving by substitution | Number of solutions |
---|---|

A value for $x$ and $y$ is determined. | One solution |

An identity is found, such as $2=2.$ | Infinitely many solutions |

A contradiction is found, such as $2 =3.$ | No solution |

$2y=-x+22$

Substitute$y=5x$

$2⋅5x=-x+22$

MultiplyMultiply

$10x=-x+22$

AddEqn$LHS+x=RHS+x$

$11x=22$

DivEqn$LHS/11=RHS/11$

$x=2$

c

${2y−8x=4y−2=4x (I)(II) $ Now we want to find out how many solutions the system has. We will do that by first solving the system using the substitution method. After that we will with help of this table determine how many solutions the system has.

Result of solving by substitution | Number of solutions |
---|---|

A value for $x$ and $y$ is determined. | One solution |

An identity is found, such as $2=2.$ | Infinitely many solutions |

A contradiction is found, such as $2 =3.$ | No solution |