maths.free › Algebra › Systems of equations

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.

例題 · 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}

ステップバイステップ

  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.

答えを表示
x = 2,\; y = 1

今度はあなたの番 問題を選択し、タイプしたり描いたりしてください。 すべてのステップは絵で、答えはあなたが尋ねるまで隠されています。

自分の解答を記録しよう

無料アカウントでは、すべてのレッスンにノートを追加し、完成したことの記録、解いた問題を一つの場所に保存し、このページについて質問できる先生を追加します。数学自体はログインしたかどうかに関係なく誰でも利用できます。

登録 ログイン

ここで使用された記号

どの記号をタップしても、定義、画像、それぞれの文字の意味が表示されます。

どうやって: 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.

よくある質問

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.

More in Algebra

Linear equationsQuadratic equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value