Integrated Math 1: Substitute and Solve: Mastering Systems of Equations

Solving Systems of Equations with Substitution 🧮

Solving systems of equations using substitution is a powerful algebraic technique. It's particularly useful when one equation is already solved for one variable, or can be easily manipulated to be. Here's a breakdown:

The Substitution Method: A Step-by-Step Guide 🪜

1. Solve for a Variable: Choose one equation and solve it for one of its variables. For example, solve for $x$ or $y$.
2. Substitute: Substitute the expression you found in step 1 into the other equation. This will result in a single equation with only one variable.
3. Solve: Solve the new equation for the remaining variable.
4. Back-Substitute: Substitute the value you found in step 3 back into either of the original equations (or the equation you solved in step 1) to find the value of the other variable.
5. Check: Check your solution by plugging both values into both original equations to ensure they are true.

Example 1: A Simple Case 🧩

Consider the following system of equations:

y = 2x + 1
x + y = 7

1. The first equation is already solved for $y$.
2. Substitute $2x + 1$ for $y$ in the second equation: $x + (2x + 1) = 7$.
3. Solve for $x$: $3x + 1 = 7 \Rightarrow 3x = 6 \Rightarrow x = 2$.
4. Substitute $x = 2$ back into $y = 2x + 1$: $y = 2(2) + 1 = 5$.
5. Check: $2 + 5 = 7$ (True) and $5 = 2(2) + 1$ (True).

Therefore, the solution is $x = 2$ and $y = 5$, or the ordered pair $(2, 5)$.

Example 2: When You Need to Solve First ✍️

Consider the following system of equations:

2x + y = 8
x - y = 1

1. Solve the second equation for $x$: $x = y + 1$.
2. Substitute $y + 1$ for $x$ in the first equation: $2(y + 1) + y = 8$.
3. Solve for $y$: $2y + 2 + y = 8 \Rightarrow 3y = 6 \Rightarrow y = 2$.
4. Substitute $y = 2$ back into $x = y + 1$: $x = 2 + 1 = 3$.
5. Check: $2(3) + 2 = 8$ (True) and $3 - 2 = 1$ (True).

Therefore, the solution is $x = 3$ and $y = 2$, or the ordered pair $(3, 2)$.

Example 3: Dealing with Fractions ➗

Consider the following system of equations:

x/2 + y = 5
x - y = 1

1. Solve the second equation for $x$: $x = y + 1$.
2. Substitute $y + 1$ for $x$ in the first equation: $(y + 1)/2 + y = 5$.
3. Solve for $y$: Multiply both sides by 2: $y + 1 + 2y = 10 \Rightarrow 3y = 9 \Rightarrow y = 3$.
4. Substitute $y = 3$ back into $x = y + 1$: $x = 3 + 1 = 4$.
5. Check: $4/2 + 3 = 5$ (True) and $4 - 3 = 1$ (True).

Therefore, the solution is $x = 4$ and $y = 3$, or the ordered pair $(4, 3)$.

When Substitution is Most Useful 🤔

Substitution shines when:

• One of the equations is already solved for a variable.
• One of the variables has a coefficient of 1 or -1, making it easy to isolate.

Practice Problems ✍️

Solve the following systems of equations using substitution:

1. $y = 3x - 2$, $x + 2y = 8$
2. $x - y = 4$, $2x + 3y = 7$
3. $a = 4b + 1$, $3a - 2b = 17$

By following these steps and practicing, you'll master solving systems of equations using substitution in no time! 🎉