Solving Systems of Equations: Substitution of Fractions in 2x-y/2 + -3y+x/5 = 7

Solve the following equations for x and y using the substitution method:

{2x−y2+−3y+x5=7−y−x8−5y+x6=4 \begin{cases} \frac{2x-y}{2}+\frac{-3y+x}{5}=7 \\ \frac{-y-x}{8}-\frac{5y+x}{6}=4 \end{cases}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve the system of equations
00:16 Multiply by 10 to get rid of fractions
00:36 Open parentheses properly, multiply by each term
00:56 Collect terms
01:06 Isolate X
01:20 This is the value of X, we'll substitute it in the second equation to find Y
01:47 Multiply by 24 to get rid of fractions
02:06 Open parentheses properly, multiply by each term
02:34 Collect terms
02:48 Substitute the X value we found to find Y
03:00 Multiply by 12 to get rid of the fraction
03:23 Open parentheses properly, multiply by each term
03:38 Isolate Y
04:10 This is the value of Y
04:21 Now let's substitute the Y value to find X
05:13 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the following equations for x and y using the substitution method:

{2x−y2+−3y+x5=7−y−x8−5y+x6=4 \begin{cases} \frac{2x-y}{2}+\frac{-3y+x}{5}=7 \\ \frac{-y-x}{8}-\frac{5y+x}{6}=4 \end{cases}

2

Step-by-step solution

To solve the system of equations using substitution:

Step 1: Simplify each equation.

First equation:

2x−y2+−3y+x5=7\frac{2x - y}{2} + \frac{-3y + x}{5} = 7

Multiply by 10 to eliminate denominators:

(2x−y)×5+(−3y+x)×2=70(2x - y) \times 5 + (-3y + x) \times 2 = 70 10x−5y−6y+2x=7010x - 5y - 6y + 2x = 70 12x−11y=70(1)12x - 11y = 70\quad\text{(1)}

Second equation:

−y−x8−5y+x6=4\frac{-y - x}{8} - \frac{5y + x}{6} = 4

Multiply by 24 to eliminate denominators:

(−y−x)×3−(5y+x)×4=96( -y - x) \times 3 - (5y + x) \times 4 = 96 −3y−3x−20y−4x=96-3y - 3x - 20y - 4x = 96 −7x−23y=96(2)-7x - 23y = 96\quad\text{(2)}

Step 2: Solve the first equation for xx:

12x=70+11y12x = 70 + 11y x=70+11y12(3)x = \frac{70 + 11y}{12}\quad\text{(3)}

Step 3: Substitute Equation (3) into Equation (2):

−7(70+11y12)−23y=96-7\left(\frac{70 + 11y}{12}\right) - 23y = 96

Clear while multiplying by 12:

−7(70+11y)−276y=1152-7(70 + 11y) - 276y = 1152 −490−77y−276y=1152-490 - 77y - 276y = 1152 −353y=1642-353y = 1642 y=−4.65y = -4.65

Step 4: Substitute y=−4.65y = -4.65 back into Equation (3) to find xx:

x=70+11(−4.65)12x = \frac{70 + 11(-4.65)}{12} x=70−51.1512x = \frac{70 - 51.15}{12} x=18.8512x = \frac{18.85}{12} x=1.57x = 1.57

Therefore, the solution to the problem is x=1.57 x = 1.57 , y=−4.65 y = -4.65 .

3

Final Answer

x=1.57,y=−4.65 x=1.57,y=-4.65

Practice Quiz

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Solve the following equations:

\( \begin{cases} 2x+y=9 \\ x=5 \end{cases} \)

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