Solving Linear Systems of Equations Using Substitution

Math Problem Statement

Use substitution to solve the systems of equations: (a) y = 5x, x + y = -6, and (b) x + 2y = 8, 3x - 4y = 4.

Let's solve these systems of equations using the substitution method.

y = 5x \\ x + y = -6 \end{cases}$$ #### **Step 1: Substitute $$y = 5x$$ into the second equation** $$x + 5x = -6$$ #### **Step 2: Solve for $$x$$** $$6x = -6$$ $$x = -1$$ #### **Step 3: Substitute $$x = -1$$ into $$y = 5x$$** $$y = 5(-1) = -5$$ #### **Solution:** $$\boxed{(-1, -5)}$$ --- ### **Problem b** $$\begin{cases} x + 2y = 8 \\ 3x - 4y = 4 \end{cases}$$ #### **Step 1: Solve for $$x$$ in terms of $$y$$ from the first equation** $$x = 8 - 2y$$ #### **Step 2: Substitute $$x = 8 - 2y$$ into the second equation** $$3(8 - 2y) - 4y = 4$$ #### **Step 3: Expand** $$24 - 6y - 4y = 4$$ $$24 - 10y = 4$$ #### **Step 4: Solve for $$y$$** $$-10y = -20$$ $$y = 2$$ #### **Step 5: Substitute $$y = 2$$ into $$x = 8 - 2y$$** $$x = 8 - 2(2) = 8 - 4 = 4$$ #### **Solution:** $$\boxed{(4,2)}$$ Would you like any further explanation? ### **Related Questions** 1. How does the substitution method compare to the elimination method in solving systems of equations? 2. Can a system of equations have no solution or infinitely many solutions? How can you tell? 3. What are some real-life applications of solving systems of equations? 4. How would the solutions change if we modified one of the equations slightly? 5. Can you solve the same systems using the elimination method and verify the answers? ### **Tip:** When using substitution, always isolate one variable first to make substitution easier! 🚀

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Linear Equations
Substitution Method
Systems of Equations

y = mx + c (linear equation)
Substitution into a second equation to solve systems

Properties of Equality

Grades 8-10

Solving a System of Equations Using Substitution Method

Solving Systems of Linear Equations by Substitution: Example with x - 5y = 10 and -2x - 6y = -4

Solving a System of Equations Using Substitution Method: 6x + 2y = 4 and 2x + 3y = 5

Solve the System of Equations: y = 3x - 5 and y = -6x + 4

Solving a System of Linear Equations with Substitution: x=4, x + y = -6, 4x - 3y + 2z = 26