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Systems of equations

Substitution and elimination, and what the intersecting lines mean.

Two equations in two unknowns are two lines; the solution is where they cross. Substitution solves one equation for a variable and pushes it into the other. For bigger systems the same idea becomes row reduction on the augmented matrix.

tusaale · x + y = 3, x - y = 1

Algebra · Systems-Of-Equations

Solve the system x + y = 3

\begin{cases}x + y = 3 \\ x - y = 1\end{cases}

tallaabo tallaabo

  1. \begin{cases}x + y = 3 \\ x - y = 1\end{cases}

    The system has 2 equations in 2 unknowns (x, y).

  2. x = 3 - y

    From the first equation, express x in terms of y.

  3. 2 - 2 y = 0

    Substitute that into the second equation, leaving one equation in y.

  4. y = 1

    Solve for y.

  5. x = 3 - y = 2

    Back-substitute to get x.

Jawaabta muuji
x = 2,\; y = 1

Haddaad Dooro dhibaato, ama qoraal ama sawirkaaga. Talaab kasta, sawir, jawaabta qarsoon ilaa aad weydiiso.

Ku hay shaqadaada

A xisaab bilaash ah oo ku daray qoraalada waxbarasho kasta, diiwaanka waxa aad dhamaatay, dhibaatooyinka aad xal u tahay meel kaliya, iyo macallinka aad weydiisan kartaa oo ku saabsan bogga. Xisaabta oo keliya waa u furan yahay qof kasta, la diiwaan geliyey ama aan.

Ku soo biir Ku soo gal

Calaamadaha la isticmaalay halkan

Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

Sida loo: Systems of equations

  1. The system has 2 equations in 2 unknowns (x, y).
  2. From the first equation, express x in terms of y.
  3. Substitute that into the second equation, leaving one equation in y.
  4. Solve for y.
  5. Back-substitute to get x.

Su'aalaha dadku waydiiyaan

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

In ka badan Algebra

Linear equationsQuadratic equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value