Solve systems of equations by substitution

Substitution solves a system of two linear equations by recognizing that equivalent expressions for the same variable can be set equal: isolate a variable in one equation, replace it in the other, and determine the remaining variable before finding the ordered-pair solution. The solution satisfies both relationships and represents the intersection of their lines; checking it in both original equations helps distinguish one solution, no solution, and infinitely many solutions. The scope is limited to familiar two-variable linear systems, not nonlinear systems or matrix-based methods.

Detailed Explanation: Solve systems of equations by substitution

Substitution works by replacing one variable with an equivalent expression.

Consider the system:

\begin{aligned} 2x+y&=11\\ x-y&=1 \end{aligned}

1. Isolate one variable

Use the first equation and solve for $y$ :

2x+y=11

Subtract $2x$ from both sides:

y=11-2x

Now we know that $y$ is equivalent to $11-2x$ .

2. Substitute into the other equation

The second equation is:

x-y=1

Replace $y$ with $11-2x$ :

x-(11-2x)=1

3. Solve for the remaining variable

Distribute the negative sign:

x-11+2x=1

Combine like terms:

3x-11=1

Add $11$ to both sides:

3x=12

Divide by $3$ :

x=4

4. Find the other variable

Substitute $x=4$ into $y=11-2x$ :

y=11-2(4)

y=11-8=3

So the solution is:

\boxed{(4,3)}

5. Check the solution

Substitute $x=4$ and $y=3$ into both original equations:

2(4)+3=11

4-3=1

Both equations are true, so $(4,3)$ is the solution. It represents the point where the two lines intersect.

Learn by doing: Solve systems of equations by substitution

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Linear Equations - Find Intersection (Decimal) - Two Linear Equations

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