Solve the System of Equations -2x+3y+4z=5 6x+7y+z=9 3x+7y+z=-9
The given problem is asking to find the values of the variables x, y, and z that satisfy all three linear equations concurrently. This is a system of linear equations with three variables, and the problem involves using mathematical methods to determine the exact values for x, y, and z that make all three equations true at the same time.
$- 2 x + 3 y + 4 z = 5$$6 x + 7 y + z = 9$$3 x + 7 y + z = - 9$
Answer
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.
