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 [latex]x,y[/latex], and [latex]z[/latex]. 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 [latex]z[/latex] and [latex]y[/latex] into equation (1) and solve for [latex]x[/latex]. 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 [latex]z[/latex] into one of the equations and solve for [latex]y[/latex]. 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, [latex]\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}[/latex]. 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 of $12,000 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 \($670\). [latex]\begin{align}−2x+4y−6z&=−18 \\ 2x−5y+5z&=17 \\ \hline −y−z&=−1\end{align}[/latex][latex]\hspace{5mm}\begin{align}&(2)\text{ multiplied by }−2\\&\left(3\right)\\&(5)\end{align}[/latex]. 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 [latex]x[/latex], we can see that we can eliminate [latex]x[/latex] by adding equation (1) to equation (2). 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