Solve the System of Equations -2x+3y+4z=5 6x+7y+z=9 3x+7y+z=-9

Solution:

Step 1: Isolate the variable $z$ in all equations.

• For the first equation, add $2x-3y$ to both sides: $4z=5+2x-3y$.

• For the second equation, subtract $6x+7y$ from both sides: $z=9-6x-7y$.

• For the third equation, subtract $3x+7y$ from both sides: $z=-9-3x-7y$.

Step 2: Express $z$ from the second equation and substitute in the others.

• From the second equation, we have $z=9-6x-7y$.

• Substitute $z$ in the first equation: $-2x+3y+4(9-6x-7y)=5$.

• Substitute $z$ in the third equation: $3x+7y+(9-6x-7y)=-9$.

Step 3: Simplify the equations after substitution.

• Expand and simplify the first equation: $-2x+3y+36-24x-28y=5$.

• Combine like terms: $-26x-25y+36=5$.

• Simplify the third equation: $3x+7y+9-6x-7y=-9$.

• Combine like terms: $-3x+9=-9$.

Step 4: Solve the simplified equations for $x$ and $y$.

• Solve for $x$: $-3x+9=-9$ implies $-3x=-18$, so $x=6$.

• Substitute $x=6$ into the first simplified equation and solve for $y$: $-26(6)-25y+36=5$ implies $-25y-120=5$, so $y=-5$.

Step 5: Substitute $x$ and $y$ back into the equation for $z$.

• Substitute $x=6$ and $y=-5$ into $z=9-6x-7y$: $z=9-6(6)-7(-5)$.

• Simplify to find $z$: $z=9-36+35$, so $z=8$.

Step 6: Write the solution as an ordered triple.

• The solution is $(6,-5,8)$.

Step 7: Present the solution in different forms.

• Point form: $(6,-5,8)$.

• Equation form: $x=6,y=-5,z=8$.

Solution:"The system of equations has a unique solution: $(6,-5,8)$."

Knowledge Notes:

• A system of linear equations can be solved by isolating one variable and substituting it into the other equations.

• The distributive property is used to expand expressions like $a(b+c)$ into $ab+ac$.

• Combining like terms involves adding or subtracting coefficients of the same variable.

• To solve for a variable, you can use inverse operations such as addition/subtraction or multiplication/division to isolate the variable on one side of the equation.

• An ordered triple $(x,y,z)$ represents the solution to a system of three equations in three variables.

• The solution to a system of equations can be expressed in point form, which represents a point in three-dimensional space, or in equation form, which lists the values of each variable.