Answer:
The equations have one solution at (5, -5).
Step-by-step explanation:
We are given a system of equations:
![\displaystyle{\left \{ {{-2x+5y=-35} \atop {7x+2y=25}} \right.}](https://tex.z-dn.net/?f=%5Cdisplaystyle%7B%5Cleft%20%5C%7B%20%7B%7B-2x%2B5y%3D-35%7D%20%5Catop%20%7B7x%2B2y%3D25%7D%7D%20%5Cright.%7D)
This system of equations can be solved in three different ways:
- Graphing the equations (method used)
- Substituting values into the equations
- Eliminating variables from the equations
<u>Graphing the Equations</u>
We need to solve each equation and place it in slope-intercept form first. Slope-intercept form is
.
Equation 1 is
. We need to isolate y.
![\displaystyle{-2x + 5y = -35}\\\\5y = 2x - 35\\\\\frac{5y}{5} = \frac{2x - 35}{5}\\\\y = \frac{2}{5}x - 7](https://tex.z-dn.net/?f=%5Cdisplaystyle%7B-2x%20%2B%205y%20%3D%20-35%7D%5C%5C%5C%5C5y%20%3D%202x%20-%2035%5C%5C%5C%5C%5Cfrac%7B5y%7D%7B5%7D%20%3D%20%5Cfrac%7B2x%20-%2035%7D%7B5%7D%5C%5C%5C%5Cy%20%3D%20%5Cfrac%7B2%7D%7B5%7Dx%20-%207)
Equation 1 is now
.
Equation 2 also needs y to be isolated.
![\displaystyle{7x+2y=25}\\\\2y=-7x+25\\\\\frac{2y}{2}=\frac{-7x+25}{2}\\\\y = -\frac{7}{2}x + \frac{25}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%7B7x%2B2y%3D25%7D%5C%5C%5C%5C2y%3D-7x%2B25%5C%5C%5C%5C%5Cfrac%7B2y%7D%7B2%7D%3D%5Cfrac%7B-7x%2B25%7D%7B2%7D%5C%5C%5C%5Cy%20%3D%20-%5Cfrac%7B7%7D%7B2%7Dx%20%2B%20%5Cfrac%7B25%7D%7B2%7D)
Equation 2 is now
.
Now, we can graph both of these using a data table and plotting points on the graph. If the two lines intersect at a point, this is a solution for the system of equations.
The table below has unsolved y-values - we need to insert the value of x and solve for y and input these values in the table.
![\begin{array}{|c|c|} \cline{1-2} \textbf{x} & \textbf{y} \\ \cline{1-2} 0 & a \\ \cline{1-2} 1 & b \\ \cline{1-2} 2 & c \\ \cline{1-2} 3 & d \\ \cline{1-2} 4 & e \\ \cline{1-2} 5 & f \\ \cline{1-2} \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7C%7D%20%5Ccline%7B1-2%7D%20%5Ctextbf%7Bx%7D%20%26%20%5Ctextbf%7By%7D%20%5C%5C%20%5Ccline%7B1-2%7D%200%20%26%20a%20%5C%5C%20%5Ccline%7B1-2%7D%201%20%26%20b%20%5C%5C%20%5Ccline%7B1-2%7D%202%20%26%20c%20%5C%5C%20%5Ccline%7B1-2%7D%203%20%26%20d%20%5C%5C%20%5Ccline%7B1-2%7D%204%20%26%20e%20%5C%5C%20%5Ccline%7B1-2%7D%205%20%26%20f%20%5C%5C%20%5Ccline%7B1-2%7D%20%5Cend%7Barray%7D)
![\bullet \ \text{For x = 0,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%200%2C%7D)
![\displaystyle{y = \frac{2}{5}(0) - 7}\\\\y = 0 - 7\\\\y = -7](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%20%3D%20%5Cfrac%7B2%7D%7B5%7D%280%29%20-%207%7D%5C%5C%5C%5Cy%20%3D%200%20-%207%5C%5C%5C%5Cy%20%3D%20-7)
![\bullet \ \text{For x = 1,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%201%2C%7D)
![\displaystyle{y=\frac{2}{5}(1)-7}\\\\y=\frac{2}{5}-7\\\\y = -\frac{33}{5}](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D%5Cfrac%7B2%7D%7B5%7D%281%29-7%7D%5C%5C%5C%5Cy%3D%5Cfrac%7B2%7D%7B5%7D-7%5C%5C%5C%5Cy%20%3D%20-%5Cfrac%7B33%7D%7B5%7D)
![\bullet \ \text{For x = 2,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%202%2C%7D)
![\displaystyle{y=\frac{2}{5}(2)-7}\\\\y = \frac{4}{5}-7\\\\y = -\frac{31}{5}](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D%5Cfrac%7B2%7D%7B5%7D%282%29-7%7D%5C%5C%5C%5Cy%20%3D%20%5Cfrac%7B4%7D%7B5%7D-7%5C%5C%5C%5Cy%20%3D%20-%5Cfrac%7B31%7D%7B5%7D)
![\bullet \ \text{For x = 3,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%203%2C%7D)
![\displaystyle{y=\frac{2}{5}(3)-7}\\\\y= \frac{6}{5}-7\\\\y=-\frac{29}{5}](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D%5Cfrac%7B2%7D%7B5%7D%283%29-7%7D%5C%5C%5C%5Cy%3D%20%5Cfrac%7B6%7D%7B5%7D-7%5C%5C%5C%5Cy%3D-%5Cfrac%7B29%7D%7B5%7D)
![\bullet \ \text{For x = 4,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%204%2C%7D)
![\displaystyle{y=\frac{2}{5}(4)-7}\\\\y = \frac{8}{5}-7\\\\y=-\frac{27}{5}](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D%5Cfrac%7B2%7D%7B5%7D%284%29-7%7D%5C%5C%5C%5Cy%20%3D%20%5Cfrac%7B8%7D%7B5%7D-7%5C%5C%5C%5Cy%3D-%5Cfrac%7B27%7D%7B5%7D)
![\bullet \ \text{For x = 5,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%205%2C%7D)
![\displaystyle{y=\frac{2}{5}(5)-7}\\\\y=2-7\\\\y=-5](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D%5Cfrac%7B2%7D%7B5%7D%285%29-7%7D%5C%5C%5C%5Cy%3D2-7%5C%5C%5C%5Cy%3D-5)
Now, we can place these values in our table.
![\begin{array}{|c|c|} \cline{1-2} \textbf{x} & \textbf{y} \\ \cline{1-2} 0 & -7 \\ \cline{1-2} 1 & -33/5 \\ \cline{1-2} 2 & -31/5 \\ \cline{1-2} 3 & -29/5 \\ \cline{1-2} 4 & -27/5 \\ \cline{1-2} 5 & -5 \\ \cline{1-2} \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7C%7D%20%5Ccline%7B1-2%7D%20%5Ctextbf%7Bx%7D%20%26%20%5Ctextbf%7By%7D%20%5C%5C%20%5Ccline%7B1-2%7D%200%20%26%20-7%20%5C%5C%20%5Ccline%7B1-2%7D%201%20%26%20-33%2F5%20%5C%5C%20%5Ccline%7B1-2%7D%202%20%26%20-31%2F5%20%5C%5C%20%5Ccline%7B1-2%7D%203%20%26%20-29%2F5%20%5C%5C%20%5Ccline%7B1-2%7D%204%20%26%20-27%2F5%20%5C%5C%20%5Ccline%7B1-2%7D%205%20%26%20-5%20%5C%5C%20%5Ccline%7B1-2%7D%20%5Cend%7Barray%7D)
As we can see in our table, the rate of decrease is
. In case we need to determine more values, we can easily either replace x with a new value in the equation or just subtract
from the previous value.
For Equation 2, we need to use the same process. Equation 2 has been resolved to be
. Therefore, we just use the same process as before to solve for the values.
![\bullet \ \text{For x = 0,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%200%2C%7D)
![\displaystyle{y=-\frac{7}{2}(0)+\frac{25}{2}}\\\\y = 0 + \frac{25}{2}\\\\y = \frac{25}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D-%5Cfrac%7B7%7D%7B2%7D%280%29%2B%5Cfrac%7B25%7D%7B2%7D%7D%5C%5C%5C%5Cy%20%3D%200%20%2B%20%5Cfrac%7B25%7D%7B2%7D%5C%5C%5C%5Cy%20%3D%20%5Cfrac%7B25%7D%7B2%7D)
![\bullet \ \text{For x = 1,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%201%2C%7D)
![\displaystyle{y=-\frac{7}{2}(1)+\frac{25}{2}}\\\\y = -\frac{7}{2} + \frac{25}{2}\\\\y = 9](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D-%5Cfrac%7B7%7D%7B2%7D%281%29%2B%5Cfrac%7B25%7D%7B2%7D%7D%5C%5C%5C%5Cy%20%3D%20-%5Cfrac%7B7%7D%7B2%7D%20%2B%20%5Cfrac%7B25%7D%7B2%7D%5C%5C%5C%5Cy%20%3D%209)
![\bullet \ \text{For x = 2,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%202%2C%7D)
![\displaystyle{y=-\frac{7}{2}(2)+\frac{25}{2}}\\\\y = -7+\frac{25}{2}\\\\y = \frac{11}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D-%5Cfrac%7B7%7D%7B2%7D%282%29%2B%5Cfrac%7B25%7D%7B2%7D%7D%5C%5C%5C%5Cy%20%3D%20-7%2B%5Cfrac%7B25%7D%7B2%7D%5C%5C%5C%5Cy%20%3D%20%5Cfrac%7B11%7D%7B2%7D)
![\bullet \ \text{For x = 3,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%203%2C%7D)
![\displaystyle{y=-\frac{7}{2}(3)+\frac{25}{2}}\\\\y = -\frac{21}{2}+\frac{25}{2}\\\\y = 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D-%5Cfrac%7B7%7D%7B2%7D%283%29%2B%5Cfrac%7B25%7D%7B2%7D%7D%5C%5C%5C%5Cy%20%3D%20-%5Cfrac%7B21%7D%7B2%7D%2B%5Cfrac%7B25%7D%7B2%7D%5C%5C%5C%5Cy%20%3D%202)
![\bullet \ \text{For x = 4,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%204%2C%7D)
![\displaystyle{y=-\frac{7}{2}(4)+\frac{25}{2}}\\\\y=-14+\frac{25}{2}\\\\y = -\frac{3}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D-%5Cfrac%7B7%7D%7B2%7D%284%29%2B%5Cfrac%7B25%7D%7B2%7D%7D%5C%5C%5C%5Cy%3D-14%2B%5Cfrac%7B25%7D%7B2%7D%5C%5C%5C%5Cy%20%3D%20-%5Cfrac%7B3%7D%7B2%7D)
![\bullet \ \text{For x = 5,}](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5Ctext%7BFor%20x%20%3D%205%2C%7D)
![\displaystyle{y=-\frac{7}{2}(5)+\frac{25}{2}}\\\\y = -\frac{35}{2}+\frac{25}{2}\\\\y = -5](https://tex.z-dn.net/?f=%5Cdisplaystyle%7By%3D-%5Cfrac%7B7%7D%7B2%7D%285%29%2B%5Cfrac%7B25%7D%7B2%7D%7D%5C%5C%5C%5Cy%20%3D%20-%5Cfrac%7B35%7D%7B2%7D%2B%5Cfrac%7B25%7D%7B2%7D%5C%5C%5C%5Cy%20%3D%20-5)
And now, we place these values into the table.
![\begin{array}{|c|c|} \cline{1-2} \textbf{x} & \textbf{y} \\ \cline{1-2} 0 & 25/2 \\ \cline{1-2} 1 & 9 \\ \cline{1-2} 2 & 11/2 \\ \cline{1-2} 3 & 2 \\ \cline{1-2} 4 & -3/2 \\ \cline{1-2} 5 & -5 \\ \cline{1-2} \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7C%7D%20%5Ccline%7B1-2%7D%20%5Ctextbf%7Bx%7D%20%26%20%5Ctextbf%7By%7D%20%5C%5C%20%5Ccline%7B1-2%7D%200%20%26%2025%2F2%20%5C%5C%20%5Ccline%7B1-2%7D%201%20%26%209%20%5C%5C%20%5Ccline%7B1-2%7D%202%20%26%2011%2F2%20%5C%5C%20%5Ccline%7B1-2%7D%203%20%26%202%20%5C%5C%20%5Ccline%7B1-2%7D%204%20%26%20-3%2F2%20%5C%5C%20%5Ccline%7B1-2%7D%205%20%26%20-5%20%5C%5C%20%5Ccline%7B1-2%7D%20%5Cend%7Barray%7D)
When we compare our two tables, we can see that we have one similarity - the points are the same at x = 5.
Equation 1 Equation 2
![\begin{array}{|c|c|} \cline{1-2} \textbf{x} & \textbf{y} \\ \cline{1-2} 0 & 25/2 \\ \cline{1-2} 1 & 9 \\ \cline{1-2} 2 & 11/2 \\ \cline{1-2} 3 & 2 \\ \cline{1-2} 4 & -3/2 \\ \cline{1-2} 5 & -5 \\ \cline{1-2} \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7C%7D%20%5Ccline%7B1-2%7D%20%5Ctextbf%7Bx%7D%20%26%20%5Ctextbf%7By%7D%20%5C%5C%20%5Ccline%7B1-2%7D%200%20%26%2025%2F2%20%5C%5C%20%5Ccline%7B1-2%7D%201%20%26%209%20%5C%5C%20%5Ccline%7B1-2%7D%202%20%26%2011%2F2%20%5C%5C%20%5Ccline%7B1-2%7D%203%20%26%202%20%5C%5C%20%5Ccline%7B1-2%7D%204%20%26%20-3%2F2%20%5C%5C%20%5Ccline%7B1-2%7D%205%20%26%20-5%20%5C%5C%20%5Ccline%7B1-2%7D%20%5Cend%7Barray%7D)
Therefore, using this data, we have one solution at (5, -5).