Solve systems of two linear equations by substitution and elimination, interpret the solution as an intersection point, and recognise systems with no solution or infinitely many solutions (Algebra).

What this topic is asking

A system of two linear equations asks for the values of two variables that satisfy both equations at once, which graphically is the intersection point of two lines. The ACT expects you to solve such systems quickly by substitution or elimination, and to recognise when two lines are parallel (no solution) or identical (infinitely many solutions).

The two methods

Pick whichever is faster for the system in front of you.

Solving by elimination

Solving by substitution

When one equation gives a variable directly, substitution is fastest. For $y = 2x - 1$ and $3x + y = 14$, replace $y$: $3x + (2x - 1) = 14$, so $5x - 1 = 14$, $5x = 15$, $x = 3$, and then $y = 2(3) - 1 = 5$. The solution is $(3, 5)$. Substituting the expression for the variable, not a number, is the key step.

No solution and infinitely many

The number of solutions matches how the two lines sit:

• One solution: lines have different slopes and cross once.
• No solution: lines are parallel (same slope, different intercepts); the algebra produces a false statement like $1 = -3$.
• Infinitely many: lines are identical (one equation is a multiple of the other); the algebra produces a true statement like $0 = 0$.

You can often see this at a glance by comparing slopes: rewrite both in $y = mx + b$ form and compare $m$ and $b$.

Word problems as systems

Many ACT word problems hide a system: two unknowns with two conditions, such as "tickets cost \8 for adults and \5 for children; 200 tickets sold for \$1,375". Let$a$and$c$be the counts, write$a + c = 200$and$8a + 5c = 1375$, then eliminate or substitute. Translating the two sentences into two equations is the whole challenge; the solving is routine. Define your variables clearly and keep units consistent.

Reading a system from a graph

An ACT question may show two lines and ask for the solution, or describe a graph in words. The solution is simply the point where the lines cross, read off the axes. If the lines are drawn parallel, there is no solution; if they coincide, every point is a solution. This graphical view is a useful check on algebra: after solving $2x + 3y = 12$ and $x - y = 1$ to get $(3, 2)$, you can confirm that $(3, 2)$ lies on both lines by substituting it back into each equation. Connecting the algebra (a solved pair) to the geometry (an intersection point) is exactly the kind of cross-representation the ACT rewards.

Try this

Q1. Solve $x + 2y = 7$ and $x = 3$. [1 point]

• Cue. Substitute $x = 3$: $3 + 2y = 7$, so $y = 2$. Solution $(3, 2)$.

Q2. How many solutions does $2x + y = 5$ and $4x + 2y = 10$ have? [1 point]

• Cue. The second is twice the first, so the lines are identical: infinitely many solutions.