Algebra Challenge: Solve the System of Equations 5y + 3x = 15 and -2y - 4x = -34

Question

5y+3x=15 5y+3x=15

2y4x=34 -2y-4x=-34

Video Solution

Step-by-Step Solution

To solve the given system of equations using the elimination method, we proceed as follows:

  • Step 1: Align the equations to eliminate one variable.
    We have: 5y+3xamp;=15(Equation 1)2y4xamp;=34(Equation 2) \begin{aligned} 5y + 3x &= 15 \quad \text{(Equation 1)} \\ -2y - 4x &= -34 \quad \text{(Equation 2)} \end{aligned}

  • Step 2: Make the coefficients of y y equal in magnitude by manipulating the equations.
    Multiply Equation 1 by 2 and Equation 2 by 5:
    (2)(5y+3x)amp;=21510y+6x=30(5)(2y4x)amp;=5(34)10y20x=170 \begin{aligned} (2) \cdot (5y + 3x) &= 2 \cdot 15 \quad \Rightarrow \quad 10y + 6x = 30 \\ (5) \cdot (-2y - 4x) &= 5 \cdot (-34) \quad \Rightarrow \quad -10y - 20x = -170 \end{aligned}

  • Step 3: Add the modified equations to eliminate y y .
    (10y+6x)+(10y20x)amp;=30+(170)0y14xamp;=14014xamp;=140 \begin{aligned} (10y + 6x) + (-10y - 20x) &= 30 + (-170) \\ 0y - 14x &= -140 \\ -14x &= -140 \end{aligned}

  • Step 4: Solve for x x .
    Dividing by 14-14:
    xamp;=14014xamp;=10 \begin{aligned} x &= \frac{-140}{-14} \\ x &= 10 \end{aligned}

  • Step 5: Substitute x=10 x = 10 back into one of the original equations to solve for y y .
    Using Equation 1:
    5y+3(10)amp;=155y+30amp;=155yamp;=15305yamp;=15yamp;=155yamp;=3 \begin{aligned} 5y + 3(10) &= 15 \\ 5y + 30 &= 15 \\ 5y &= 15 - 30 \\ 5y &= -15 \\ y &= \frac{-15}{5} \\ y &= -3 \end{aligned}

Therefore, the solution to the system of equations is x=10 x = 10 and y=3 y = -3 .

Answer

x=10,y=3 x=10,y=-3