Systems of Linear Equations

Cancel terms when the question asks for a sum

Given $2x+3y=17$ and $4x-3y=7$, add the equations. The $y$ terms cancel, leaving $6x=24$, so $x=4$. Substitute into the first equation to get $y=3$.

Before solving, read the requested quantity. If the equations are $x+y=8$ and $x-y=2$, adding them gives $x=5$, but a question asking for $x+y$ already has its answer in the first equation.

Substitute an isolated variable

For $y=2x+1$ and $x+y=10$, replace $y$ in the second equation: $x+(2x+1)=10$. Thus $x=3$ and $y=7$. Keep parentheses around the substituted expression when it is multiplied or subtracted.

A graphing solution finds the intersection of the two lines. Read both coordinates and check the required one; the graph's horizontal coordinate is $x$, even if the question asks for $y$.

Tell no solution from infinitely many solutions

The system $2x+4y=8$ and $x+2y=5$ has no solution. Doubling the second equation would give $2x+4y=10$, which contradicts the first. The lines have equal slopes and different intercepts.

Replacing the second constant with 4 makes the equations describe the same line, giving infinitely many solutions. All corresponding coefficients and the constant must have the same multiplier.

Check both constraints in a word problem

Suppose adult tickets cost 8 dollars, child tickets cost 5 dollars, and 10 tickets bring in 65 dollars. Write $a+c=10$ and $8a+5c=65$. Substitution gives $a=5$ and $c=5$.

The count equation checks the total number of tickets; the revenue equation checks the total money. Checking only one equation would allow many incorrect pairs.

Check your understanding

Sample systems of linear equations questions

Practice this skill

1. Choose elimination or substitution and say which expression made that choice useful.
2. Check the resulting pair in both original equations.
3. Include coefficient questions about zero, one, and infinitely many solutions.

Print the systems of linear equations worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.