代数のレッスン：代数の問題の解き方

代数の問題はどれも、いくつかの操作に行き着きます：簡単にする、両辺に同じことをする、確かめる。このレッスンでは、Sumzo の数学エンジンが答えを確認した例題で、それぞれの操作を学べます。

18 件の記事

5 分で読めます
Equations: what they are and how to solve them
What an equation is, what its solution means, and how to solve equations step by step — from one-step equations to fractions and two variables.
1 分で読めます
Solving linear equations
Use inverse operations to get the unknown on its own, keep the equation balanced, and always check your answer.
2 分で読めます
Quadratic equations: how to solve them
Factorising, completing the square, the quadratic formula and the discriminant — and how to tell which method to use.
1 分で読めます
Slope and the equation of a line
What slope measures, how to find it from two points, and how to write a line as y = mx + b.
4 分で読めます
How to simplify radicals (simplest radical form)
How to simplify radicals by taking out perfect-square factors, what simplest radical form means, and how to simplify radical expressions — with checked examples.
2 分で読めます
Rational equations: how to solve them
What a rational equation is, how to solve rational equations by multiplying by the lowest common denominator, and why every answer must be checked.
3 分で読めます
Factoring polynomials: a step-by-step guide
The steps for factoring polynomials — common factor first, then two, three or four terms — with worked examples, each checked by multiplying back out.
3 分で読めます
Solving inequalities: how to solve and graph them
Solving inequalities like equations — with one extra rule — and showing the answer on a number line, in interval notation and as a shaded region.
2 分で読めます
What is a solution in math? Examples with one, none or many
A solution is a value that makes an equation true. How to check one, and examples with one solution, no solution, all real numbers and two solutions.
2 分で読めます
How to solve a word problem: 5 steps that work
A five-step method for solving word problems, the key words that turn into maths, and worked examples — ages, consecutive numbers, discounts and tickets.
2 分で読めます
How to graph an equation: lines, parabolas and more
Three ways to graph an equation — a table of values, slope and intercept, and key points of a parabola — with worked examples and how to match a graph to its equation.
3 分で読めます
Functions: how to graph a function, domain, range and f(g(x))
Function notation, how to graph a function from its formula, how to find its domain and range, and how composite functions such as f(g(x)) work.
2 分で読めます
Polynomials: how to simplify, expand and solve them
What a polynomial is, how to simplify and expand polynomials, and how to solve polynomial equations by factoring — with checked examples.
2 分で読めます
Rational expressions: simplify, multiply, divide, add and subtract
What rational expressions are, their excluded values, and how to simplify, multiply, divide, add and subtract them — then solve rational equations.
2 分で読めます
How to solve literal equations (rearranging formulas)
A literal equation has several letters. Solving it means isolating one of them — the same moves as an ordinary equation, with letters instead of numbers.
1 分で読めます
How to simplify expressions and complex fractions
How to simplify the following kinds of expression: like terms, brackets, exponents and complex fractions (a fraction inside a fraction), each with checked examples.
1 分で読めます
Find x: how to solve for x in equations, angles and triangles
The questions that say “find x” — equations, angles on a line, triangle angles, Pythagoras and similar shapes — and the one idea behind all of them.
2 分で読めます
Systems of equations and graphing systems of linear inequalities
How to find the solution to a system of equations three ways, then how to graph a system of inequalities: boundary lines, shading and the overlap.

代数の問題への取り組み方

何が問われているかを読みましょう：式を簡単にするのか、ある文字について方程式を解くのか？
それぞれの辺を簡単にします：かっこを外し、同類項をまとめます。
計算を逆の順で戻します — まず足し算と引き算、次に掛け算と割り算 — 両辺に同じことをします。
答えを元の問題に代入して確かめます。

代数の例題

$4(x - 2) = 2x + 6$ を解きます。展開： $4x - 8 = 2x + 6$ 。 $2x$ を引く： $2x - 8 = 6$ 。8 を足す： $2x = 14$ 、よって $x = 7$ 。

確認： $4(7 - 2) = 20$ 、 $2(7) + 6 = 20$ 。両辺が一致するので $x = 7$ は正しいです。

どの代数レッスンが必要？

未知数が 1 つの方程式：方程式と一次方程式から始めましょう。
問題に $x^2$ がある：二次方程式、次に多項式の因数分解。
$x$ を含む分数：分数式と分数方程式。
グラフ、 $f(x)$ 、 $y =$ ：方程式のグラフ、関数、傾き。
記号ではなく文章：文章題の解き方。

よくある質問

自分の問題について代数の手助けはどこで得られる？

Sumzo の代数計算機に入力してください。すべての手順を表示し、数学エンジンが答えを確認します。ここのレッスンでは、その手順の裏にある方法を学べます。

代数の答えはどう確かめる？

答えを元の方程式に代入します。両辺が等しくなれば正解です — そうでなければ、途中式で方程式が変わってしまった最初の行を探しましょう。

代数でまず何を学ぶべき？

式を簡単にすることと、1 段階・2 段階の方程式を解くことです。二次方程式、関数、連立方程式など、その後に学ぶものはすべてこれが土台です。