# system of equations problems 3 variables

So the general solution is $$\left(x,\dfrac{5}{2}x,\dfrac{3}{2}x\right)$$. Solve the resulting two-by-two system. How much did John invest in each type of fund? Step 1. 12. Then, we write the three equations as a system. We will get another equation with the variables x and y and name this equation as (5). A system of equations in three variables is inconsistent if no solution exists. This leaves two equations with two variables--one equation from each pair. STEP Substitute the values found in Step 2 into one of the original equations and solve for the remaining variable. We will check each equation by substituting in the values of the ordered triple for $x,y$, and $z$. Find the equation of the circle that passes through the points , , and Solution. Step 4. 14. You discover a store that has all jeans for $25 and all dresses for$50. A system of equations is a set of equations with the same variables. So, let’s first do the multiplication. These two steps will eliminate the variable $$x$$. It makes no difference which equation and which variable you choose. Then, back-substitute the values for $z$ and $y$ into equation (1) and solve for $x$. The final equation $$0=2$$ is a contradiction, so we conclude that the system of equations in inconsistent and, therefore, has no solution. \begin{align*} x+y+z &= 2 \nonumber \\[4pt] 6x−4y+5z &= 31 \nonumber \\[4pt] 5x+2y+2z &= 13 \nonumber \end{align*} \nonumber. The second step is multiplying equation (1) by $$−2$$ and adding the result to equation (3). A system of equations is a set of one or more equations involving a number of variables. There are other ways to begin to solve this system, such as multiplying equation (3) by $$−2$$, and adding it to equation (1). In this system, each plane intersects the other two, but not at the same location. For more information contact us at [email protected] or check out our status page at https://status.libretexts.org. \begin{align} −4x−2y+6z =0 & (1) \;\;\;\;\; \text{multiplied by }−2 \nonumber \\[4pt] \underline{4x+2y−6z=0} & (2) \nonumber \\[4pt] 0=0& \nonumber \end{align} \nonumber. Example $$\PageIndex{5}$$: Finding the Solution to a Dependent System of Equations. The third equation shows that the total amount of interest earned from each fund equals 670. Infinite number of solutions of the form $$(x,4x−11,−5x+18)$$. Equation 3) 3x - 2y – 4z = 18 Systems of Three Equations. This also shows why there are more “exceptions,” or degenerate systems, to the general rule of 3 equations being enough for 3 variables. We back-substitute the expression for $z$ into one of the equations and solve for $y$. Choosing one equation from each new system, we obtain the upper triangular form: \begin{align} x−2y+3z=9 \; &(1) \nonumber \\[4pt] y+2z =3 \; &(4) \nonumber \\[4pt] z=2 \; &(6) \nonumber \end{align} \nonumber. Call the changed equations … At the er40f the Thus, \begin{align}x+y+z=12{,}000 \hspace{5mm} \left(1\right) \\ -y+z=4{,}000 \hspace{5mm} \left(2\right) \\ 3x+4y+7z=67{,}000 \hspace{5mm} \left(3\right) \end{align}. Tim wants to buy a used printer. Use the answers from Step 4 and substitute into any equation involving the remaining variable. 2. John received an inheritance of12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. Graphically, an infinite number of solutions represents a line or coincident plane that serves as the intersection of three planes in space. The total interest earned in one year was $$670$$. \begin{align}−2x+4y−6z&=−18 \\ 2x−5y+5z&=17 \\ \hline −y−z&=−1\end{align}\hspace{5mm}\begin{align}&(2)\text{ multiplied by }−2\\&\left(3\right)\\&(5)\end{align}. Any point where two walls and the floor meet represents the intersection of three planes. In the problem posed at the beginning of the section, John invested his inheritance of $$12,000$$ in three different funds: part in a money-market fund paying $$3\%$$ interest annually; part in municipal bonds paying $$4\%$$ annually; and the rest in mutual funds paying $$7\%$$ annually. This will be the sample equation used through out the instructions: Equation 1) x – 6y – 2z = -8. Then, we multiply equation (4) by 2 and add it to equation (5). Looking at the coefficients of $x$, we can see that we can eliminate $x$ by adding equation (1) to equation (2). A solution to a system of three equations in three variables $\left(x,y,z\right),\text{}$ is called an ordered triple. 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