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SAT Systems of Equations Practice

A system of linear equations describes two conditions that must hold at the same time. A solution is an ordered pair that satisfies both equations, or the point where their lines intersect.

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The key ideas

  • A system's solution is the point that satisfies both equations.
  • Substitution is efficient when one variable is already isolated.
  • Matching lines have infinitely many solutions; parallel lines have none.

Choose substitution or elimination

Use substitution when an equation already isolates x or y: replace that variable in the other equation with its equal expression. Use elimination when adding or subtracting the equations can cancel a variable. If you multiply an equation to align coefficients, multiply every term on both sides.

Read the final question carefully. If it asks for x + y rather than the ordered pair, combining the equations may give the requested expression directly. Check the result against both original conditions.

One solution, no solution, or infinitely many?

Different slopes mean the lines intersect once. Equal slopes and different intercepts mean parallel lines and no solution. Equivalent equations describe the same line and infinitely many solutions. In algebra, cancellation to a false statement signals no solution; cancellation to an identity signals infinitely many.

Worked example

Solve y=2x+1y = 2x + 1 and y=x+4y = x + 4.

  1. Set the two expressions for y equal: 2x+1=x+42x + 1 = x + 4.
  2. Solve x=3x = 3, then substitute to get y=7y = 7.

Answer: (3, 7)

A common mistake to avoid

Finding one variable and forgetting to substitute back for the other.

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Curriculum and sources

The key ideas and worked example come from PrepSAT's reviewed lesson curriculum. These College Board resources define the SAT scope; PrepSAT is an independent practice resource and is not affiliated with College Board.